WikiJournal Preprints/Cut the coordinates! (or Vector Analysis Done Fast)

Introduction

Sheldon Axler, in his essay "Down with determinants!" (1995) and his ensuing book Linear Algebra Done Right (4th Ed., 2023–), does not entirely eliminate determinants, but introduces them as late as possible and then exploits them for what he calls their "main reasonable use in undergraduate mathematics", namely the change-of-variables formula for multiple integrals. Here I treat coordinates in vector analysis somewhat as Axler treats determinants in linear algebra: I introduce coordinates as late as possible, and then exploit them in unconventionally rigorous derivations of vector-analytic identities from (e.g.) vector-algebraic identities. But I contrast with Axler in at least two ways. First, as my subtitle suggests, I have no intention of expanding my paper into a book. Brevity is of the essence. Second, while one may well avoid determinants in numerical  linear algebra, one can hardly avoid coordinates in numerical vector analysis! So I cannot extend the coordinate-minimizing path into computation. But I can extend it up to the threshold by expressing the operators of vector analysis in general coordinates and orthogonal coordinates, leaving it for others to specialize the coordinates and compute with them. On the way, I can satisfy readers who need the concepts of vector analysis for theoretical purposes, and who would rather read a paper than a book. Readers who stay to the end of the paper will get a more general treatment of coordinates than is offered by a typical book-length introduction to vector analysis. In the meantime, coordinates won't needlessly get in the way.

The cost of coordinates

Mathematicians define a "vector" as a member of a vector space, which is a set whose members satisfy certain basic rules of algebra (called the vector-space axioms) with respect to another set called a field (e.g., the real numbers), which has its own basic rules of algebra (the field axioms), and whose members are called "scalars". Physicists are more fussy. They typically want a "vector" to be not only a member of a vector space, but also a first-order tensor : a "tensor", meaning that it exists independently of any coordinate system with which it might be specified; and "first-order" (or "first-degree", or "first-rank"), meaning that it is specified by a one-dimensional array of numbers. Similarly, a 2nd-order tensor is specified by a 2-dimensional array (a matrix), and a 3rd-order by a 3-dimensional array, and so on. Hence they want a "scalar", which is specified by a single number (a zero-dimensional array), to be a zero-order tensor. In "vector analysis", we are greatly interested in applications to physical situations, and accordingly take the physicists' view on what constitutes a vector or a scalar.

So, for our purposes, defining a quantity by three components in (say) a Cartesian coordinate system is not enough to make it a vector, and defining a quantity as a real function of a list of coordinates is not enough to make it a scalar, because we still need to show that the quantity has an independent existence. One way to do this is to show that its coordinate representation behaves appropriately when the coordinate system is changed. Independent existence of a quantity means that its coordinate representation changes so as to compensate for the change in the coordinate system. But independent existence of an operator means that its expression in one coordinate system (with the operand[s] and the result in that system) gives the same result as the corresponding expression in another coordinate system.

Here we circumvent these complications by the most obvious route: by initially defining things without coordinates. If, having defined something without coordinates, we then need to represent it with coordinates, we can choose the coordinate system for convenience rather than generality.

The limitations of limits

In the branch of pure mathematics known as analysis, there is a thing called a limit, whereby for every positive ϵ  there exists a positive δ such that if some increment is less than δ, some error is less than ϵ. In the branch of applied mathematics known as continuum mechanics, there is a thing called reality, whereby if the increment is less than some positive δ, the assumption of a continuum becomes ridiculous, so that the error cannot be made less than an arbitrary ϵ. Yet vector "analysis" (together with higher-order tensors) is typically studied with the intention of applying it to some form of "continuum" mechanics, such as the modeling of elasticity, plasticity, fluid flow, or (widening the net) electrodynamics of ordinary matter; in short, it is studied with the intention of conveniently forgetting that, on a sufficiently small scale, matter is lumpy.Template:Efn One might therefore submit that to express the principles of vector analysis in the language of limits is to strain at a gnat and swallow a camel. Here I avoid that camel by referring to elements of length or area or volume, each of which is small enough to allow some quantity or quantities to be considered uniform within it, but, for the same reason, large enough to allow such local averaging of the said quantity or quantities as is necessary to tune out the lumpiness.

We shall see bigger camels, where well-known authors define or misdefine a vector operator and then derive identities by treating it like an ordinary vector quantity. These I also avoid.

Prerequisites

I assume that the reader is familiar with the algebra and geometry of vectors in 3D space, including the dot-product, the cross-product, and the scalar triple product, their geometric meanings, their expressions in Cartesian coordinates, and the identity

Template:Big

which we call the "expansion" of the vector triple product. I further assume that the reader can generalize the concept of a derivative, so as to differentiate a vector with respect to a scalar, e.g.

${𝐫}^{\prime }(t)=\frac{d𝐫}{dt}=\underset{h\to 0}{\lim }\frac{𝐫(t+h)-𝐫(t)}{h},$

or so as to differentiate a function of several independent variables "partially" w.r.t. one of them while the others are held constant, e.g.

$\frac{\partial }{\partial y}\psi (x,y,z)=\underset{h\to 0}{\lim }\frac{\psi (x,y+h,z)-\psi (x,y,z)}{h}.$

But in view of the above remarks on limits, I also expect the reader to be tolerant of an argument like this: In a short timeTemplate:Mvar, let the vectors Template:Math andTemplate:Math change by Template:Math andTemplate:Math respectively. Then

$\begin{array}{cc}\frac{d}{dt}(𝐫\times 𝐩)& =\frac{(𝐫+d𝐫)\times (𝐩+d𝐩)-𝐫\times 𝐩}{dt}\\ & =\frac{𝐫\times d𝐩+d𝐫\times 𝐩}{dt}[𝗇𝖾𝗀𝗅𝖾𝖼𝗍𝗂𝗇𝗀d𝐫\times d𝐩]\\ & =𝐫\times \frac{d𝐩}{dt}+\frac{d𝐫}{dt}\times 𝐩\\ & =𝐫\times \dot{𝐩}+\dot{𝐫}\times 𝐩,\end{array}$

where, as always, the orders of the cross-products matter.Template:Efn Differentiation of a dot-product behaves similarly, except that the orders don't matter; and if Template:Math, where Template:Mvar is a scalar and Template:Math is a vector, then

$\dot{𝐩}=m\dot{𝐯}+\dot{m}𝐯.$

Or an argument like this:  If$z=f(x,y)$, then

$\begin{array}{cc}\frac{{\partial }^{2}z}{\partial x\partial y}& =\frac{\partial }{\partial x}\frac{\partial }{\partial y}f(x,y)\\ & =\frac{\partial }{\partial x}\frac{f(x,y+dy)-f(x,y)}{dy}\\ & =\frac{\frac{f(x+dx,y+dy)-f(x+dx,y)}{dy}-\frac{f(x,y+dy)-f(x,y)}{dy}}{dx}\\ & =\frac{\frac{f(x+dx,y+dy)-f(x,y+dy)}{dx}-\frac{f(x+dx,y)-f(x,y)}{dx}}{dy}\\ & =\frac{\partial }{\partial y}\frac{f(x+dx,y)-f(x,y)}{dx}\\ & =\frac{\partial }{\partial y}\frac{\partial }{\partial x}f(x,y)=\frac{{\partial }^{2}z}{\partial y\partial x};\end{array}$

that is, we can switch the order of differentiation in a "mixed" partial derivative. IfTemplate:Mvar is an abbreviation for Template:Mvar, etc., this rule can be written in operational terms as

Template:Big

More generally, if Template:Mvar is an abbreviation for Template:Mvar  where  Template:Math  the rule becomes

Template:Big

These generalizations of differentiation, however, do not go beyond differentiation w.r.t. real variables, some of which are scalars, and some of which are coordinates. Vector analysis involves quantities that may be loosely described as derivatives w.r.t. a vector—usually the position vector.

Closed-surface integrals per unit volume

The term field, mentioned above in the context of algebraic axioms, has an alternative meaning: if Template:Math is the position vector, a scalar field is a scalar-valued function ofTemplate:Math and a vector field is a vector-valued function ofTemplate:Math; both may also depend on time. These are the functions of which we want "derivatives" w.r.t. the vectorTemplate:Math.

In this section I introduce four such derivatives—the gradient, the curl, the divergence, and the Laplacian —in a way that will seem unremarkable to those readers who aren't already familiar with them, but idiosyncratic to those who are. The gradient is commonly introduced in connection with a curve and its endpoints, the curl in connection with a surface segment and its enclosing curve, the divergence in connection with a volume and its enclosing surface, and the Laplacian as a composite of two of the above, initially applicable only to a scalar field. Here I introduce all four in connection with a volume and its enclosing surface; and I introduce the Laplacian as a concept in its own right, equally applicable to a scalar or vector  field, and only later relate it to the others. My initial definitions of the gradient, the curl, and the Laplacian, although not novel, are usually thought to be more advanced than the common ones—in spite of being conceptually simpler, and in spite of being obvious variations on the same theme.

Instant integral theorems (with a caveat)

Let Template:Mvar be a volume (3D region) enclosed by a surface Template:Mvar (a mathematical surface, not generally a physical barrier). Let Template:Math be the unit normal vector at a general point on Template:Mvar, pointing out ofTemplate:Mvar. Let Template:Mvar be the distance from Template:Mvar in the direction ofTemplate:Math (positive outside Template:Mvar, negative inside), and let Template:Mvar be an abbreviation forTemplate:Mvar, where the derivative—commonly called the normal derivative—is tacitly assumed to exist.

In Template:Mvar, and on Template:Mvar, let Template:Mvar be a scalar field (e.g., pressure in a fluid, or temperature), and let Template:Math be a vector field (e.g., flow velocity, or heat-flow density), and let Template:Mvar be a generic field which may be a scalar or a vector. Let a general element of the surface Template:Mvar have area Template:Mvar, and let it be small enough to allow Template:Math, Template:Mvar, Template:Math, and Template:Mvar to be considered uniform over the element. Then, for every element, the following four products are well defined: Template:NumBlk If Template:Mvar is pressure in a non-viscous fluid, the first of these products is the force exerted by the fluid in Template:Mvar  through the area Template:Mvar. The second product does not have such an obvious physical interpretation; but ifTemplate:Math is circulating clockwise about an axis directed throughTemplate:Mvar, the cross-product will be exactly tangential toTemplate:Mvar and will tend to have a component in the direction of that axis. The third product is the flux ofTemplate:Math through the surface element; ifTemplate:Math is flow velocity, the third product is the volumetric flow rate (volume per unit time) out ofTemplate:Mvar  throughTemplate:Mvar; or if Template:Math is heat-flow density, the third product is the heat transfer rate (energy per unit time) out ofTemplate:Mvar  throughTemplate:Mvar. The fourth product, by analogy with the third, might be called the flux of the normal derivative ofTemplate:Mvar through the surface element, but is equally well defined whether Template:Mvar is a scalar or a vector—or, for that matter, a matrix, or a tensor of any order, or anything else that we can differentiate w.r.t.Template:Mvar.

If we add up each of the four products over all the elements of the surface Template:Mvar, we obtain, respectively, the four surface integrals Template:NumBlk in which the double integral sign indicates that the range of integration is two-dimensional. The first surface integral takes a scalar field and yields a vector; the second takes a vector field and yields a vector; the third takes a vector field and yields a scalar; and the fourth takes (e.g.) a scalar field yielding a scalar, or a vector field yielding a vector. IfTemplate:Mvar is pressure in a non-viscous fluid, the first integral is the force exerted by the fluid in Template:Mvar  on the fluid outside Template:Mvar. The second integral may be called the skew surface integral ofTemplate:Math over Template:Mvar, or, for the reason hinted above, the circulation ofTemplate:Math over Template:Mvar.  The third integral, commonly called the flux integral (or simply the surface integral) ofTemplate:Math over Template:Mvar, is the total flux ofTemplate:Math out ofTemplate:Mvar. And the fourth integral is the surface integral of the outward normal derivative ofTemplate:Mvar.

Let the volume Template:Mvar  be divided into elements. Let a general volume element have the volume Template:Mvar and be enclosed by the surface Template:Mvar —not to be confused with the area Template:Mvar of a surface element, which may be an element ofTemplate:Mvar or ofTemplate:Mvar. Then consider what happens if, instead of evaluating each of the above surface integrals over Template:Mvar, we evaluate it over each Template:Mvar and add up the results for all the volume elements. In the interior ofTemplate:Mvar, each surface element of area Template:Mvar is on the boundary between two volume elements, for which the unit normals Template:Math at Template:Mvar, and the respective values ofTemplate:Mvar, are equal and opposite. Hence when we add up the integrals over the surfaces Template:Mvar, the contributions from the elements Template:Mvar cancel in pairs, except on the original surface Template:Mvar, so that we are left with the original integral over Template:Mvar. So, for the four surface integrals in (Template:EquationNote), we have respectively Template:NumBlk

Now comes a big "if":  if  we define the gradient ofTemplate:Mvar (pronounced "grad Template:Mvar") inside Template:Mvar  as Template:NumBlk and the curl of Template:Math inside Template:Mvar  as Template:NumBlk and the divergence of Template:Math inside Template:Mvar  as Template:NumBlk and the Laplacian of Template:Mvar inside Template:Mvar  as Template:Efn Template:NumBlk (where the letters after the equation number stand for gradient, curl, divergence, and Laplacian, respectively), then equations (Template:EquationNote) can be rewritten

$\begin{array}{cc}\underset{S}{\iint }\widehat{𝐧}pdS& =\sum _V\mathrm{\nabla }pdV,\\ \underset{S}{\iint }\widehat{𝐧}\times 𝐪dS& =\sum _V\mathrm{curl}𝐪dV,\\ \underset{S}{\iint }\widehat{𝐧}\mathbf{\cdot }𝐪dS& =\sum _V\mathrm{div}𝐪dV,\\ \underset{S}{\iint }{\partial }_n\psi dS& =\sum _V\mathrm{△}\psi dV.\end{array}$

But because each term in each sum has a factor Template:Mvar, we call the sum an integral; and because the range of integration is three-dimensional, we use a triple integral sign. Thus we obtain the following four theorems relating integrals over an enclosing surface Template:Mvar  to integrals over the enclosed volume Template:Mvar: Template:NumBlk Template:NumBlk Template:NumBlk Template:NumBlk Of the above four results, only the third (Template:EquationNote) seems to have a standard name; it is called the divergence theorem (or Gauss's theorem or, more properly, Ostrogradsky's theorem), and is indeed the best known of the four—although the other three, having been derived in parallel with it, may be said to be equally fundamental.

As each of the operators Template:Math Template:Math and Template:Math calls for an integration w.r.t. area and then a division by volume, the dimension (or unit of measurement) of the result is the dimension of the operand divided by the dimension of length, as if the operation were some sort of differentiation w.r.t. position. Moreover, in each of equations (Template:EquationNote) to (Template:EquationNote), there is a triple integral on the right but only a double integral on the left, so that each of the operators Template:Math Template:Math and Template:Math appears to compensate for a single integration. For these reasons, and for convenience, we shall describe them as differential operators. By comparison, the Template:Mathoperator in (Template:EquationNote) or (Template:EquationNote) calls for a further differentiation w.r.t.Template:Mvar; we shall therefore describe Template:Math as a 2nd-order differential operator. (An additional reason for these descriptions will emerge later.) As promised, the four definitions (Template:EquationNote) to (Template:EquationNote) are "obvious variations on the same theme" (although the fourth is somewhat less obvious than the others).

But remember the "if": Theorems (Template:EquationNote) to (Template:EquationNote) depend on definitions (Template:EquationNote) to (Template:EquationNote) and are therefore only as definite as those definitions! Equations (Template:EquationNote), without assuming anything about the shapes and relative sizes of the closed surfaces Template:Mvar (except, tacitly, that Template:Math is piecewise well-defined), indicate that the surface integrals are additive with respect to volume. But this additivity, by itself, does not guarantee that the surface integrals are shared among neighboring volume elements in proportion to their volumes, as envisaged by "definitions" (Template:EquationNote) to (Template:EquationNote). Each of these "definitions" is unambiguous if, and only if, the ratio of the surface integral toTemplate:Mvar  is insensitive to the shape and size ofTemplate:Mvar  for a sufficiently small Template:Mvar. Notice that the issue here is not whether the ratios specified in equations (Template:EquationNote) to (Template:EquationNote) are true vectors or scalars, independent of the coordinates; all of the operations needed in those equations have coordinate-free definitions. Rather, the issue is whether the resulting ratios are unambiguous notwithstanding the ambiguity of Template:Mvar, provided only that Template:Mvar is sufficiently small. That is the advertised "caveat", which must now be addressed.

In accordance with our "applied" mathematical purpose, our proofs of the unambiguity of the differential operators will rest on a few thought experiments, each of which applies an operator to a physical field, sayTemplate:Mvar, and obtains another physical field whose unambiguity is beyond dispute. The conclusion of the thought experiment is then applicable to any operand field whose mathematical properties are consistent with its interpretation as the physical fieldTemplate:Mvar; the loss of generality, if any, is only what is incurred by that interpretation.

Unambiguity of the gradient

Suppose that a fluid with density Template:Mvar (a scalar field) flows with velocityTemplate:Math (a vector field) under the influence of the internal pressure Template:Mvar (a scalar field). Then the integral in (Template:EquationNote) is the force exerted by the pressure of the fluid inside Template:Mvar on the fluid outside, so that minus the integral is the force exerted on the fluid insideTemplate:Mvar  by the pressure of the fluid outside. Dividing by Template:Mvar, we find that Template:Math, as defined by (Template:EquationNote), is the force per unit volume, due to the pressure outside the volume. If this is the only force per unit volume acting on the volume (e.g., because the fluid is non-viscous and in a weightless environment, and the volume element is not in contact with the container), then it is equal to the acceleration times the mass per unit volume; that is, Template:NumBlk Now provided that the left side of this equation is locally continuous, it can be considered uniform inside the small Template:Mvar, so that the left side is unambiguous, whence  Template:Mathis also unambiguous. If there are additional forces on the fluid element, e.g. due to gravity and⧸or viscosity, then Template:Math is not the sole contribution to density-times-acceleration, but is still the contribution due to pressure, which is still unambiguous.

By showing the unambiguity of definition (Template:EquationNote), we have confirmed theorem (Template:EquationNote). In the process we have seen that the volume-based definition of the gradient is useful for the modeling of fluids, and intuitive in that it formalizes the common notion that a pressure "gradient" gives rise to a force.

Unambiguity of the divergence

In the aforesaid fluid, in a short timeTemplate:Mvar, the volume that flows out of fixed closed surface Template:Mvar  through a fixed surface element of area Template:Mvar  is Template:Math  (i.e., the displacement normal to the surface element, times the area).  Multiplying this by density and integrating over Template:Mvar, we find that the mass flowing out ofTemplate:Mvar  in timeTemplate:Mvar is  $\underset{\delta S}{\iint }\rho 𝐯dt\cdot \widehat{𝐧}dS$.  Dividing this by Template:Mvar, and then by Template:Mvar, we get the rate of reduction of density inside Template:Mvar; that is,

$\frac{1}{dV}\underset{\delta S}{\iint }\widehat{𝐧}\cdot \rho 𝐯dS=-\frac{\partial \rho }{\partial t},$

where the derivative w.r.t. time is evaluated at a fixed location (because Template:Mvar is fixed), and is therefore written as a partial derivative (because other variables on which Template:Mvar might depend—namely the coordinates—are held constant). Provided that the right-hand side is locally continuous, it can be considered uniform inside Template:Mvar and is therefore unambiguous, so that the left side is likewise unambiguous. But the left side is simplyTemplate:Math  as defined by (Template:EquationNote),Template:Efn which is therefore also unambiguous, confirming theorem (Template:EquationNote). In short, the divergence operator is that which maps Template:Math to the rate of reduction of density at a fixed point: Template:NumBlk This result, which expresses conservation of mass, is a form of the so-called equation of continuity.

The partial derivative Template:Mvar in (Template:EquationNote) must be distinguished from the material derivative Template:Mvar, which is evaluated at a point that moves with the fluid.Template:Efn [Similarly, Template:Math in (Template:EquationNote) is the material acceleration, because it is the acceleration of the mobile mass—not of a fixed point! ]  To re-derive the equation of continuity in terms of the material derivative, the volume Template:Math which flows out throughTemplate:Mvar in timeTemplate:Mvar (as above), is integrated over Template:Mvar to obtain the increase in volume of the mass initially contained in Template:Mvar. Dividing this by the mass, Template:Mvar, gives the increase in specific volume Template:Math of that mass, and then dividing by Template:Mvar gives the rate of change of specific volume; that is,

$\frac{1}{\rho dV}\underset{\delta S}{\iint }\widehat{𝐧}\cdot 𝐯dS=\frac{d}{dt}\left({\rho }^{-1}\right)=-{\rho }^{-2}\frac{d\rho }{dt}.$

Multiplying by Template:Math and comparing the left side with (Template:EquationNote), we obtain Template:NumBlk Whereas (Template:EquationNote) shows that Template:Math is unambiguous, (Template:EquationNote) shows that Template:Math is unambiguous (provided that the right-hand sides are locally continuous). In accordance with the everyday meaning of "divergence", (Template:EquationNote) also shows that Template:Math is positive if the fluid is expanding (Template:Mvardecreasing), negative if it is contracting (Template:Mvarincreasing), and zero if it is incompressible. In the last case, the equation of continuity reduces to Template:NumBlk

For incompressible flow, any tubular surface tangential to the flow velocity, and consequently with no flow in or out of the "tube", has the same volumetric flow rate across all cross-sections of the "tube", as if the surface were the wall of a pipe full of liquid (except that the surface is not necessarily stationary). Accordingly, a vector field with zero divergence is described as solenoidal (from the Greek word for "pipe"). More generally, a solenoidal vector field has the property that for any tubular surface tangential to the field, the flux integrals across any two cross-sections of the "tube" are the same—because otherwise there would be a net flux integral out of the closed surface comprising the two cross-sections and any segment of tube between them, in which case, by the divergence theorem (Template:EquationNote), the divergence would have to be non-zero somewhere inside, contrary to (Template:EquationNote).

Unambiguity of the curl (and gradient)

The unambiguity of the curl (Template:EquationNote) follows from the unambiguity of the divergence. Taking dot-products of (Template:EquationNote) with an arbitrary constant vector Template:Math we get

$\begin{array}{cc}𝐛\cdot \mathrm{curl}𝐪& =𝐛\cdot \frac{1}{dV}\underset{\delta S}{\iint }\widehat{𝐧}\times 𝐪dS\\ & =\frac{1}{dV}\underset{\delta S}{\iint }𝐛\cdot \widehat{𝐧}\times 𝐪dS\\ & =\frac{1}{dV}\underset{\delta S}{\iint }\widehat{𝐧}\cdot 𝐪\times 𝐛dS;\end{array}$

that is, by (Template:EquationNote), Template:NumBlk (The parentheses around  Template:Math  on the right, although helpful because of the spacing, are not strictly necessary, because the alternative binding would be Template:Math, which is a scalar, whose cross-product with the vector Template:Math is not defined. And the left-hand expression does not need parentheses, because it can only mean the dot-product of a curl with the vector Template:Math; it cannot mean the curl of a dot-product, because the curl of a scalar field is not defined.) This result (Template:EquationNote) is an identity if the vector Template:Math is independent of location, so that it can be taken inside or outside the surface integral; thus Template:Math may be a uniform vector field, and may be time-dependent. If we make Template:Math a unit vector, the left side of the identity is the (scalar) component of Template:Math in the direction ofTemplate:Math and the right side is unambiguous. Thus the curl is unambiguous because its component in any direction is unambiguous. This confirms theorem (Template:EquationNote).

Similarly, the unambiguity of the divergence implies the unambiguity of the gradient. Starting with (Template:EquationNote), taking dot-products with an arbitrary uniform vector Template:Math and proceeding as above, we obtain Template:NumBlk (The left-hand side does not need parentheses, because it can only mean the dot-product of a gradient with the vector Template:Math; it cannot mean the gradient of the dot-product of a scalar field with a vector field, because that dot-product would not be defined.) If we make Template:Math a unit vector, this result (Template:EquationNote) says that the (scalar) component ofTemplate:Math in the direction ofTemplate:Math is given by the right-hand side, which again is unambiguous. So here we have a second explanation of the unambiguity of the gradient: like the curl, it is unambiguous because its component in any direction is unambiguous.

We might well ask what happens if we take cross-products with Template:Math on the left, instead of dot-products. If we start with (Template:EquationNote), the process is straightforward: in the end we can switch the order of the cross-product on the left, and change the sign on the right, obtaining Template:NumBlk (Again no parentheses are needed.) If we start with (Template:EquationNote) instead, and take Template:Math inside the integral, we get a vector triple product to expand, which leads to

$𝐛\times \mathrm{curl}𝐪=\frac{1}{dV}\underset{\delta S}{\iint }\widehat{𝐧}𝐛\mathbf{\cdot }𝐪dS-\frac{1}{dV}\underset{\delta S}{\iint }𝐛\mathbf{\cdot }\widehat{𝐧}𝐪dS,$

in which the first term on the right is simply  Template:Math  (the gradient of the dot-product). The second term is more problematic. If we had a scalar Template:Mvar instead of the vector Template:Math we could take Template:Math outside the second integral, so that the second term would be (minus) Template:Math. This suggests that the actual second term should be (minus) Template:Math.  Shall we therefore adopt the second term (without the sign) as the definition ofTemplate:Math for a vector Template:Math (treating Template:Math as an operator), and write Template:NumBlk The proposal would be open to the objection that  Template:Math  had been defined only for uniformTemplate:Math whereas  Template:Math (for scalarTemplate:Mvar) is defined whether Template:Math is uniform or not.  So, for the moment, let us put (Template:EquationNote) aside and run with (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote).

Another meaning of the gradient

Let Template:Math be a unit vector in a given direction, and let Template:Mvar be a parameter measuring distance (arc length) along a path in that direction. By equation (Template:EquationNote) and definition (Template:EquationNote), we have

$\mathrm{\nabla }p\cdot \widehat{𝐬}=\mathrm{div}p\widehat{𝐬}=\frac{1}{dV}\underset{\delta S}{\iint }\widehat{𝐧}\cdot p\widehat{𝐬}dS,$

where, by the unambiguity of the divergence, the shape of the closed surface Template:Mvar enclosing Template:Mvar  can be chosen for convenience. So let Template:Mvar be a right cylinder with cross-sectional area Template:Mvar  and perpendicular height Template:Mvar with the path passing perpendicularly through the end-faces at parameter-values Template:Mvar and Template:Mvar where the outward unit normal Template:Math consequently takes the values Template:Math and Template:Math respectively. And let the cross-sectional dimensions be small compared with Template:Mvar  so that the values ofTemplate:Mvar at the end-faces, say Template:Mvar and Template:Mvar, can be taken to be the same as where the end-faces cut the path. Then  Template:Mvar, and the surface integral overTemplate:Mvar includes only the contributions from the end-faces (because Template:Math is perpendicular to Template:Math elsewhere); those contributions are respectively  $-\widehat{𝐬}\cdot p\widehat{𝐬}\alpha$ and  $\widehat{𝐬}\cdot (p+dp)\widehat{𝐬}\alpha ,$  i.e.  $-p\alpha$ and $(p+dp)\alpha$.  With these substitutions the above equation becomes

$\begin{array}{cc}\mathrm{\nabla }p\cdot \widehat{𝐬}& =\frac{1}{\alpha ds}(-p\alpha +(p+dp)\alpha )\\ & =\frac{p+dp-p}{ds}=\frac{\partial p}{\partial s};\end{array}$

that is, Template:NumBlk where the right-hand side, commonly called the directional derivative ofTemplate:Mvar in the Template:Math direction, is the derivative ofTemplate:Mvar w.r.t. distance in that direction. Although (Template:EquationNote) has been obtained by taking that direction as fixed, the equality is evidently maintained if Template:Mvar measures arc length along any path tangential  toTemplate:Math at the point of interest.

Equation (Template:EquationNote) is an alternative definition of the gradient: it says that the gradient of$p$ is the vector whose scalar component in any direction is the directional derivative of$p$ in that direction. For real$p$, this component has its maximum, namely Template:Math in the direction ofTemplate:Math; thus the gradient of$p$ is the vector whose direction is that in which the derivative of$p$ w.r.t. distance is a maximum, and whose magnitude is that maximum. This is the usual conceptual definition of the gradient. Sometimes it is convenient to work directly from this definition. For example, in Cartesian coordinates Template:Math if a scalar field is given by Template:Mvar its gradient is obviously the unit vector in the direction of the Template:Mvaraxis, usually called Template:Math; that is, Template:Math. Similarly, if  Template:Math  is the position vector, then Template:Math.

If  Template:Math is tangential  to a level surface ofTemplate:Mvar (a surface of constantTemplate:Mvar), then Template:Mvar  in that direction is zero, in which case (Template:EquationNote) says that Template:Math (if not zero) is orthogonal toTemplate:Math.  So$\mathrm{\nabla }p$ is orthogonal to the surfaces of constant$p$ (as we would expect, having just shown that the direction ofTemplate:Math is that in which Template:Mvar varies most steeply). This result leads to a method of finding a vector normal to a curved surface at a given point: if the equation of the surface is  Template:Math  where Template:Mathis the position vector and Template:Mvaris a constant (possibly zero), a suitable vector is Template:Math  evaluated at the given point.

If Template:Mvar is uniform —that is, if it has no spatial variation—then its derivative w.r.t. distance in every direction is zero; that is, the component ofTemplate:Math in every direction is zero, so that Template:Math must be the zero vector. In short, the gradient of a uniform scalar field is zero. Conversely, if Template:Mvar is not uniform, there must be some location and some direction in which its derivative w.r.t. distance, if defined at all, is non-zero, so that its gradient, if defined at all, is also non-zero. Thus a scalar field with zero gradient in some region is uniform in that region.

Unambiguity of the Laplacian

Armed with our new definition of the gradient (Template:EquationNote), we can revisit our definition of the Laplacian (Template:EquationNote). IfTemplate:Mvar is a scalar field, then, by (Template:EquationNote),  ${\partial }_n\psi$ can be replaced by $\mathrm{\nabla }\psi \cdot \widehat{𝐧}$ in (Template:EquationNote), which then becomes Template:NumBlk that is, by definition (Template:EquationNote), Template:NumBlk So the Laplacian of a scalar field is the divergence of the gradient. This is the usual introductory definition of the Laplacian—and on its face is applicable only in the case of a scalar field. The unambiguity of the Laplacian, in this case, follows from the unambiguity of the divergence and the gradient.

If, on the contrary, Template:Mvar in definition (Template:EquationNote) is a vector field, then we can again take dot-products with a uniform vector Template:Math obtaining

$(\mathrm{△}\psi )\cdot 𝐛=\frac{1}{dV}\underset{\delta S}{\iint }{\partial }_n(\psi \cdot 𝐛)dS.$

If we make Template:Math a unit vector, this says that the scalar component of the Laplacian of a vector field, in any direction, is the Laplacian of the scalar component of that vector field in that direction. As we have just established that the latter is unambiguous, so is the former.

But the unambiguity of the Laplacian can be generalized further. If

Template:Big

where each ${\phi }_i$ is a scalar field, and each Template:Mvar is a constant, and the counter Template:Mvar ranges from (say) 1 toTemplate:Mvar, then it is clear from (Template:EquationNote) that Template:NumBlk In words, this says that the Laplacian of a linear combination of fields is the same linear combination of the Laplacians of the same fields—or, more concisely, that the Laplacian is linear. I say "it is clear" because the Laplacian as defined by (Template:EquationNote) is itself a linear combination, so that (Template:EquationNote) merely asserts that we can regroup the terms of a nested linear combination; the gradient, curl, and divergence as defined by (Template:EquationNote) to (Template:EquationNote) are likewise linear. It follows from (Template:EquationNote) that the Laplacian of a linear combination of fields is unambiguous if the Laplacians of the separate fields are unambiguous. Now we have supposed that the fields ${\phi }_i$ are scalar and that the coefficients Template:Mvar are constants. But the same logic applies if the "constants" are uniform basis vectors (e.g.,Template:Math), so that the "linear combination" can represent any vector field, whence the Laplacian of any vector field is unambiguous. And the same logic applies if the "constants" are chosen as a "basis" for a space of tensors of any order, so that the Laplacian of any tensor field of that order is unambiguous, and so on. In short, the Laplacian of any field that we can express with a uniform basis is unambiguous.

The dot-del, del-cross, and del-dot operators

The gradient operator Template:Math is also called Template:Mvar.Template:Efn If it simply denotes the gradient, we tend to pronounce it "grad" in order to emphasize the result. But it can also appear in combination with other operators to give other results, and in those contexts we tend to pronounce it "del".

One such combination is "dot del"— as in "Template:Math", which we proposed for (Template:EquationNote), but did not quite manage to define satisfactorily for a vector operand. With our new definition of the gradient (Template:EquationNote), we can now make a second attempt. A general vector field Template:Math can be written Template:Math so that

$𝐪\cdot \mathrm{\nabla }\psi =|𝐪|\widehat{𝐪}\cdot \mathrm{\nabla }\psi .$

If Template:Mvar is a scalar field, we can apply (Template:EquationNote) to the right-hand side, obtaining

Template:Big

where Template:Mvar is distance in the direction ofTemplate:Math. For scalar Template:Mvar, this result is an identity between previously defined quantities. For non-scalar Template:Mvar, we have not yet defined the left-hand side, but the right-hand side is still well-defined and self-explanatory (provided that we can differentiate Template:Mvar w.r.t.Template:Mvar). So we are free to adopt Template:NumBlk where Template:Mvar is distance in the direction ofTemplate:Math as the general definition of the operator Template:Math and to interpret it as defining both a unary operator  Template:Math which operates on a generic field, and a binary operator  Template:Math which takes a (possibly uniform) vector field on the left and a generic field on the right.

For any vector field Template:Math it follows from (Template:EquationNote) that if$\psi$ is a uniform field, then$𝐪\cdot \mathrm{\nabla }\psi =0$.

For the special case in which Template:Math is a unit vector Template:Math with Template:Mvar measuring distance in the direction of  Template:Math definition (Template:EquationNote) reduces to Template:NumBlk which agrees with (Template:EquationNote) but now holds for a generic field Template:Mvar [whereas (Template:EquationNote) was for a scalar field, and was derived as a theorem based on earlier definitions]. SoTemplate:Math with a unit vector Template:Math is the directional-derivative operator on a generic field; and by (Template:EquationNote),  Template:Math is a scaled directional derivative operator on a generic field.

In particular, if  Template:Math  we have

${\partial }_n\psi =\widehat{𝐧}\cdot \mathrm{\nabla }\psi ,$

which we may substitute into the original definition of the Laplacian (Template:EquationNote) to obtain Template:NumBlk which is just (Template:EquationNote) again, except that it now holds for for a generic field.

If our general definition of the gradient (Template:EquationNote) is also taken as the general definition of the Template:Math operator, then, comparing (Template:EquationNote) with (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote), we see that

$\begin{array}{cc}\mathrm{curl}𝐪& =\mathrm{\nabla }(\times 𝐪)\\ \mathrm{div}𝐪& =\mathrm{\nabla }(\cdot 𝐪)\\ \mathrm{△}\psi & =\mathrm{\nabla }(\cdot \mathrm{\nabla }\psi ),\end{array}$

where the parentheses may seem to be required on account of the closing Template:Mvar  in (Template:EquationNote). But if we write the factor Template:Mvar before the integrand, the del operator in (Template:EquationNote) becomes

$\mathrm{\nabla }=\frac{1}{dV}\underset{\delta S}{\iint }dS\widehat{𝐧}$

—if  we insist that it is to be read as a operator looking for an operand, and not as a self-contained expression. Then, if we similarly bring forward the Template:Mvar in (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote), the respective operators become Template:NumBlk (pronounced "del cross", "del dot", and "del dot del"), of which the last is usually abbreviated asTemplate:Math  ("del squared"). These notations are ubiquitous.

Another way to obtain the Template:Math  and Template:Math  operators (but notTemplate:Math), again inspired by (Template:EquationNote), is to define Template:NumBlk where Template:Mvar  is any well-defined function that takes a vector argument. SettingTemplate:Math to Template:Math Template:Math andTemplate:Math  in (Template:EquationNote), we obtain respectively Template:Math  Template:Math and  Template:Math  as given by (Template:EquationNote) to (Template:EquationNote). But this approach has undesirable side-effects—for example, that Template:Math  becomes synonymous withTemplate:Math.  Accordingly, Chen-To Tai, on the left of (Template:EquationNote), replacesTemplate:Math with his original symbol$\mathrm{\nabla }{}^{{}_{-}},$ which he calls the "symbolic operator" or the "Template:Nowrap" or, later, the "symbolic vector" or the "dummy vector". Tai in his later works (e.g., 1994, 1995) does not tolerate cross- or dot-products involving the del operator, but does tolerate such products involving his symbolic vector (1995, pp. 50–52).

There is a misconception that the operational equivalences in (Template:EquationNote) apply only in Cartesian coordinates. Tai does not accept them even in that case. But, because these equivalences have been derived from coordinate-free definitions of the operators, they must remain valid in any coordinate system provided that they are expressed correctly—without (e.g.) inadvertently taking dependent variables inside or outside differentiations. That does not mean that they are always convenient, or easily verified, or conducive to the avoidance of error. But they sometimes make useful mnemonics; e.g., they let us rewrite identities (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) as Template:NumBlk These would be basic algebraic vector identities if  Template:Math were an ordinary vector, and one could try to derive them from the "algebraic" behavior ofTemplate:Math; but they're not, because it isn't, so we didn't !  Moreover, these simple "algebraic" rules are for a uniform Template:Math and do not of themselves tell us what to do if  Template:Math is spatially variable; for example, (Template:EquationNote) is not applicable to (Template:EquationNote).

The advection operator

Variation or transportation of a property of a medium due to motion with the medium is called advection (which, according to its Latin roots, means "carrying to"). Suppose that a medium (possibly a fluid) moves with a velocity field Template:Math in some inertial reference frame. Let Template:Mvar be a field (possibly a scalar field or a vector field) expressing some property of the medium (e.g., density, or acceleration, or stress,Template:Efn… or even Template:Mathitself). We have seen that the time-derivative ofTemplate:Mvar may be specified in two different ways: as the partial derivative Template:Mvar evaluated at a fixed point (in the chosen reference frame), or as the material derivative Template:Mvar, evaluated at a point moving at velocity Template:Math (i.e., with the medium). The difference  Template:Mvar is due to motion with the medium. To find another expression for this difference, let Template:Mvar be a parameter measuring distance along the path traveled by a particle of the medium. Then, for points along the path, the surface-plot of the small change in Template:Mvar (or any component thereof) as a function of small changes in Template:Mvar and Template:Mvar (plotted on perpendicular axes) can be taken as a plane through the origin, so that

Template:Big

that is, the change in Template:Mvar is the sum of the changes due to the change in Template:Mvar and the change in Template:Mvar. Dividing by Template:Mvar gives

Template:Big

i.e.,

Template:Big

(and the first term on the right could have been written Template:Mvar). So the second term on the right is the contribution to the material derivative due to motion with the medium; it is called the advective term, and is non-zero wherever a particle of the medium moves along a path on which Template:Mvar varies with location—even if Template:Mvar at each location is constant over time.  So the operator  Template:Math where Template:Mvar measures distance along the path, is the advection operator : it maps a property of a medium to the advective term in the time-derivative of that property. IfTemplate:Mvar is Template:Mathitself, the above result becomes

Template:Big

where the left-hand side (the material acceleration) is as given by Newton's second law, and the first term on the right (which we might call the "partial" acceleration) is the time-derivative of velocity in the chosen reference frame, and the second term on the right (the advective term) is the correction that must be added to the "partial" acceleration in order to obtain the material acceleration. This term is non-zero wherever velocity is non-zero and varies along a path, even if the velocity at each point on the path is constant over time (as when water speeds up while flowing at a constant volumetric rate into a nozzle). Paradoxically, while the material acceleration and the "partial" acceleration are apparently linear (first-degree) in Template:Math their difference (the advective term) is not. Thus the distinction between Template:Mvar and Template:Mvar  has the far-reaching implication that fluid dynamics is non-linear.

Applying (Template:EquationNote) to the last two equations, we obtain respectively Template:NumBlk and Template:NumBlk where, in each case, the second term on the right is the advective term. So the advection operator can also be written Template:Math.

When the generic Template:Mvar in (Template:EquationNote) is replaced by the density Template:Mvar, we get a relation between Template:Mvar and Template:Mvar, both of which we have seen before—in equations (Template:EquationNote) and (Template:EquationNote) above. Substituting from those equations then gives Template:NumBlk where Template:Math can be taken as a gradient since Template:Mvar is scalar. This result is in fact an identity—a product rule for the divergence—as we shall eventually confirm by another method.

Generalized volume-integral theorem

We can rewrite the fourth integral theorem (Template:EquationNote) in the "dot del" notation as Template:NumBlk Then, using notations (Template:EquationNote), we can condense all four integral theorems (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) into the single equation Template:NumBlk where the wildcard Template:Math (conveniently pronounced "star") is a generic binary operator which may be replaced by a null (direct juxtaposition of the operands) for theorem (Template:EquationNote), or a cross for (Template:EquationNote), or a dot for (Template:EquationNote), or  Template:Math for (Template:EquationNote). This single equation is a generalized volume-integral theorem, relating an integral over a volume to an integral over its enclosing surface.

Theorem (Template:EquationNote) is based on the following definitions, which have been found unambiguous:

• the gradient of a scalar field Template:Mvar is the closed-surface integral of  Template:Math per unit volume, where Template:Math is the outward unit normal;
• the curl of a vector field is the skew surface integral per unit volume, also called the surface circulation per unit volume;
• the divergence of a vector field is the outward flux integral per unit volume; and
• the Laplacian is the closed-surface integral of the outward normal derivative, per unit volume.

The gradient maps a scalar field to a vector field; the curl maps a vector field to a vector field; the divergence maps a vector field to a scalar field; and the Laplacian maps a scalar field to a scalar field, or a vector field to a vector field, etc.

The gradient of Template:Mvar, as defined above, has been shown to be also

• the vector whose (scalar) component in any direction is the directional derivative ofTemplate:Mvar in that direction (i.e. the derivative ofTemplate:Mvar w.r.t. distance in that direction), and
• the vector whose direction is that in which the directional derivative ofTemplate:Mvar is a maximum, and whose magnitude is that maximum.

Consistent with these alternative definitions of the gradient, we have defined the Template:Math operator so that  Template:Math (for a unit vector Template:Math) is the operator yielding the directional derivative in the direction of  Template:Math and we have used that notation to bring theorem (Template:EquationNote) under theorem (Template:EquationNote).

So far, we have said comparatively little about the curl. That imbalance will now be rectified.

Closed-circuit integrals per unit area

Instant integral theorems (on a condition)

Theorems (Template:EquationNote) to (Template:EquationNote) are three-dimensional: each of them relates an integral over a volume Template:Mvar  to an integral over its enclosing surfaceTemplate:Mvar. We now seek analogous two-dimensional theorems, each of which relates an integral over a surface segment to an integral around its enclosing curve. For maximum generality, the surface segment should be allowed to be curved into a third dimension.Template:Efn Theorems of this kind can be obtained as special cases of theorems (Template:EquationNote) to (Template:EquationNote) by suitably choosing Template:Mvar and Template:Mvar; this is another advantage of our "volume first" approach.

Let Template:Mvar be a surface segment enclosed by a curve Template:Mvar (a circuit or closed contour), and let Template:Mvar be a parameter measuring arc length around Template:Mvar, so that a general element ofTemplate:Mvar  has lengthTemplate:Mvar; and let a general element of the surface Template:Mvar  have area Template:Mvar. Let$\hat{\mathit{\nu }}$ be the unit normal vector at a general point on Template:Mvar, and let Template:Math be the unit tangent vector toTemplate:Mvar at a general point on Template:Mvar  in the direction of increasingTemplate:Mvar. In the original case of a surface enclosing a volume, we had to decide whether the unit normal pointed into or out of the volume (we chose the latter). In the present case of a circuit enclosing a surface segment, we have to decide whether Template:Mvar is measured clockwise or counterclockwise as seen when looking in the direction of the unit normal, and we choose clockwise. So Template:Mvaris measured clockwise about$\hat{\mathit{\nu }},$ and Template:Mvaris traversed clockwise about$\hat{\mathit{\nu }}$.

From Template:Mvar  we can construct obvious candidates for Template:Mvar andTemplate:Mvar. From every point on Template:Mvar, erect a perpendicular with a uniform small  height Template:Mvar in the direction of$\hat{\mathit{\nu }}$. Then simply let Template:Mvar be the volume occupied by all the perpendiculars, and let Template:Mvar be its enclosing surface. Thus Template:Mvar is a (generally curved) thin slab of uniform thicknessTemplate:Mvar, whose enclosing surface Template:Mvar consists of two close parallel (generally curved) broad faces connected by a perpendicular edge-face of uniform heightTemplate:Mvar; and we can treat$\hat{\mathit{\nu }}$ as a vector field  by extrapolating it perpendicularly fromTemplate:Mvar. If we can arrange for Template:Mvar to cancel out, the volumeTemplate:Mvar  will serve as a 3D representation of the surface segmentTemplate:Mvar  while the edge-face will serve as a 2D representation of the curveTemplate:Mvar, so that our four theorems will relate an integral around Template:Mvar  to an integral over Template:Mvar  provided that there is no contribution from the broad faces to the integral overTemplate:Mvar. For brevity, let us call this proviso the 2D condition.

If  the 2D condition is satisfied, an integral over the new Template:Mvar  reduces to an integral over the edge-face, on which

$dS=hdl,$

so that the cancellation ofTemplate:Mvar will leave an integral over Template:Mvar  w.r.t. length. Meanwhile, in an integral over the newTemplate:Mvar, regardless of the 2D condition, we have

$dV=hd𝛴,$

so that the cancellation ofTemplate:Mvar will leave an integral over Template:Mvar  w.r.t. area. So, substituting for Template:Mvar and Template:Mvar  in (Template:EquationNote) to (Template:EquationNote), and canceling Template:Mvar as planned, we obtain respectively Template:NumBlk Template:NumBlk Template:NumBlk Template:NumBlk all subject to the 2D condition. In each equation, the circle on the left integral sign acknowledges that the integral is around a closed loop. The unit vector Template:Math which was normal to the edge-face, is now normal to both Template:Math and$\hat{\mathit{\nu }}$; that is, Template:Math is tangential to the surface segment Template:Mvar  and projects perpendicularly outward from its bounding curve.

On the left side of (Template:EquationNote), the 2D condition is satisfied if (but not only if) Template:Math takes equal-and-opposite values at any two opposing points on opposing broad faces ofTemplate:Mvar i.e. if Template:Mvar takes the same value at such points, i.e. if Template:Mvar has a zero directional derivative normal toTemplate:Mvar i.e. if Template:Math has no component normal toTemplate:Mvar. Thus a sufficient "2D condition" for (Template:EquationNote) is the obvious one.

Skipping forward to (Template:EquationNote), we see that the 2D condition is satisfied if${\partial }_n\psi$ takes equal-and-opposite values at any two opposing points on opposing broad faces ofTemplate:Mvar i.e. if${\partial }_{\nu }\psi$ (where $\nu$ measures distance in the direction of$\hat{\mathit{\nu }}$) takes the same value at such points, i.e. if$\underset{\nu }{\overset{2}{\partial }}\psi =0$.

For (Template:EquationNote) and (Template:EquationNote), the 2D constraint can be satisfied by construction, with more useful results—as explained under the next two headings. To facilitate this process, we first make a minor adjustment to Template:Mvar andTemplate:Mvar. Noting that any curved surface segment can be approximated to any desired accuracy by a polyhedral surface enclosed by a polygon, we shall indeed consider Template:Mvar  to be a polyhedral surface made up of small planar elements, Template:Mvar  being the area of a general element, and we shall indeed consider Template:Mvar to be a polygon with short sides, Template:Mvar being the length of a general side.Template:Efn The benefit of this trick, as we shall see, is to make the unit normal $\hat{\mathit{\nu }}$ uniform over each surface element, without forcing us to treat Template:Math (or any other field) as uniform over the same element. But, as the elements ofTemplate:Mvar  can independently be made as short as we like (dividing straight sides into shorter elements if necessary!), we can still consider $\hat{\mathit{\nu }},$ Template:Math and Template:Math to be uniform over each element ofTemplate:Mvar.

Special case for the gradient

In (Template:EquationNote), the 2D condition is satisfied by$𝐪=p\hat{\mathit{\nu }}$ (where Template:Mvar is a scalar field), because then the integrand on the left is zero on the broad faces ofTemplate:Mvar, where Template:Math is parallel to$\hat{\mathit{\nu }}$. Equation (Template:EquationNote) then becomes Template:NumBlk Now on the left,  $\widehat{𝐧}\times \hat{\mathit{\nu }}=-\widehat{𝐭};$ and on the right, over each surface element, the unit normal $\hat{\mathit{\nu }}$ is uniform so that, by (Template:EquationNote),  Template:Midsize.  With these substitutions, the minus signs cancel and we get Template:NumBlk or, if we write  Template:Math  and  $𝒅\mathit{𝛴}=\hat{\mathit{\nu }}d𝛴,$ Template:NumBlk This result, although well attested in the literature, does not seem to have a name—unlike the next result.

Special case for the curl

In (Template:EquationNote), the 2D condition is satisfied if Template:Math is replaced by$\hat{\mathit{\nu }}\times 𝐪,$ because then (again) the integrand on the left is zero on the broad faces ofTemplate:Mvar, where Template:Math is parallel to$\hat{\mathit{\nu }}$. Equation (Template:EquationNote) then becomes Template:NumBlk Now on the left, the integrand can be written  $\widehat{𝐧}\times \hat{\mathit{\nu }}\cdot 𝐪=-\widehat{𝐭}\cdot 𝐪;$ and on the right,  Template:Midsize by identity (Template:EquationNote), since $\hat{\mathit{\nu }}$ is uniform over each surface element.  With these substitutions, the minus signs cancel and we get Template:NumBlk or, if we again write  Template:Math  and  $𝒅\mathit{𝛴}=\hat{\mathit{\nu }}d𝛴,$ Template:NumBlk This result—the best-known theorem relating an integral over a surface segment to an integral around its enclosing curve, and the best-known theorem involving the curl—is called Stokes' theorem or, more properly, the Kelvin–Stokes theorem, or simply the curl theorem.

The integral on the left of (Template:EquationNote) or (Template:EquationNote) is called the circulation of the vector field Template:Math around the closed curveTemplate:Mvar. So, in words, the Kelvin–Stokes theorem says that the circulation of a vector field around a closed curve is equal to the flux of the curl of that vector field through any surface spanning that closed curve.

Now let a general element of Template:Mvar (with area Template:Mvar) be enclosed by the curve Template:Mvar, traversed in the same direction as the outer curve Template:Mvar. Then, applying (Template:EquationNote) to the single element, we have

${\oint }_{\delta C}𝐪\cdot \widehat{𝐭}dl=\mathrm{curl}𝐪\cdot \hat{\mathit{\nu }}d𝛴;$

that is, Template:NumBlk where the right-hand side is simply the circulation per unit area.

Equation (Template:EquationNote) is an alternative definition of the curl: it says that the curl ofTemplate:Math is the vector whose scalar component in any direction is the circulation ofTemplate:Math per unit area of a surface whose normal points in that direction. For realTemplate:Math this component has its maximum, namely Template:Math in the direction ofTemplate:Math; thus the curl ofTemplate:Math is the vector whose direction is that which a surface must face if the circulation ofTemplate:Math per unit area of that surface is to be a maximum, and whose magnitude is that maximum. This is the usual conceptual definition of the curl.

[Notice, however, that our original volume-based definition (Template:EquationNote) is more succinct: the curl is the closed-surface circulation per unit volume, i.e. the skew surface integral per unit volume.]

It should now be clear where the curl gets its name (coined by Maxwell), and why it is also called the rotation (indeed the Template:Math operator is sometimes written "Template:Math", especially in Continental languages, in which "rot" does not have the same unfortunate everyday meaning as in English). It should be similarly unsurprising that a vector field with zero curl is described as irrotational (which one must carefully pronounce differently from "Template:Nowrap"!), and that the curl of the velocity of a medium is called the vorticity.

Animation of a non-vortex-like velocity field whose curl (like its circulation around the red loop) is non-zero due to shear.

Animation of a vortex-like velocity field whose curl is zero because the shear compensates for the rotation.

However, a field does not need to be vortex-like in order to have a non-zero curl; for example, by identity (Template:EquationNote), in Cartesian coordinates, the velocity field Template:Math has a curl equal to  Template:Math  although it describes a shearing motion rather than a rotating motion. This is understandable because if you hold a pencil between the palms of your hands and slide one palm over the other (a shearing motion), the pencil rotates. Conversely, we can have a vortex-like field whose curl is zero everywhere except on or near the axis of the vortex. For example, the Maxwell–Ampère law in magnetostatics says that  Template:Math where Template:Math is the magnetizing field and Template:Math is the current density.Template:Efn So if the current is confined to a wire, Template:Math is zero outside the wire—although, as is well known, the field lines circle the wire. The resolution of the paradox is that Template:Math gets stronger as we approach the wire, making a shearing pattern, whose effect on the curl counteracts that of the rotation.

The curl-grad and div-curl operators

We have seen from (Template:EquationNote) that the Laplacian of a scalar field is the divergence of the gradient. Four more such second-order combinations make sense, namely the curl of the gradient (of a scalar field), and the divergence of the curl, the gradient of the divergence, and the curl of the curl (of a vector field). The first two —"curl grad" and "div curl"— can now be disposed of.

Let the surface segment Template:Mvar enclosed by the curveTemplate:Mvar  be a segment of the closed surface Template:Mvar surrounding the volumeTemplate:Mvar, and let Template:Mvar expand across Template:Mvar until it engulfs{{mvar| V},} so that Template:Mvar shrinks to a point on the far side ofTemplate:Mvar. Then, in the nameless theorem (Template:EquationNote) and the Kelvin–Stokes theorem (Template:EquationNote), the integral on the left becomes zero while Template:Mvar and $\hat{\mathit{\nu }}$ on the right become Template:Mvar and Template:Math so that the theorems respectively reduce to

$\underset{S}{\iint }\widehat{𝐧}\times \mathrm{\nabla }pdS=\mathrm{𝟎}$

and

$\underset{S}{\iint }\widehat{𝐧}\cdot \mathrm{curl}𝐪dS=0.$

Applying theorem (Template:EquationNote) to the first of these two equations, and the divergence theorem (Template:EquationNote) to the second, we obtain respectively

$\underset{V}{\iiint }\mathrm{curl}\mathrm{\nabla }pdV=\mathrm{𝟎},$

and

$\underset{V}{\iiint }\mathrm{div}\mathrm{curl}𝐪dV=0.$

As the integrals vanish for any volume Template:Mvar  in which the integrands are defined, the integrands must be zero wherever they are defined; that is, Template:NumBlk and Template:NumBlk In words, the curl of the gradient is zero, and the divergence of the curl is zero; or, more concisely, any gradient is irrotational, and any curl is solenoidal.

We might well ask whether the converses are true. Is every irrotational vector field the gradient of something? And is every solenoidal vector field the curl of something? The answers are affirmative, but the proofs require more preparation.

Meanwhile we may note, as a mnemonic aid, that when the left-hand sides of the last two equations are rewritten in the del-cross and del-dot notations, they become  Template:Math  and  Template:Math respectively. The former looks like (but isn't) a cross-product of two parallel vectors, and the latter looks like (but isn't) a scalar triple product with a repeated factor, so that each expression looks like it ought to be zero (and it is). But such appearances can lead one astray, because Template:Math is an operator, not a self-contained vector quantity; for example,  Template:Math  is not identically zero, because two gradients are not necessarily parallel.

We should also note, to tie a loose end, that identity (Template:EquationNote) was to be expected from our verbal statement of the Kelvin–Stokes theorem (Template:EquationNote). That statement implies that the flux of the curl through any two surfaces spanning the same closed curve is the same. So if we make a closed surface from two spanning surfaces, the flux into one spanning surface is equal to the flux out of the other, i.e. the net flux out of the closed surface is zero, i.e. the integral of the divergence over the enclosed volume is zero; and since any simple volume in which the divergence is defined can be enclosed this way, the divergence itself (of the curl) must be zero wherever it is defined.

Change per unit length

Continuing (and concluding) the trend of reducing the number of dimensions, we now seek one-dimensional theorems, each of which relates an integral over a path to values at the endpoints of the path. For maximum generality, the path should be allowed to be curved into a second and a third dimension.

We could do this by further specializing theorems (Template:EquationNote) to (Template:EquationNote). We could take a curve Template:Math with a unit tangent vector Template:Math. At every point on Template:Math we could mount a circular disk with a uniform small area Template:Mvar centered onTemplate:Math and orthogonal to it. We could let Template:Mvar be the volume occupied by all the disks and let Template:Mvar be its enclosing surface; thus Template:Mvar would be a thin right circular cylinder, except that its axis could be curved. If we could arrange for Template:Mvar to cancel out, our four theorems would indeed be reduced to the desired form, provided that there were no contribution from the curved face of the "cylinder" to the integral overTemplate:Mvar (the "1D proviso"). But, as it turns out, this exercise yields only one case in which the "1D proviso" can be satisfied by a construction involving Template:Math and a general field, and we have already almost discovered that case by a simpler and more conventional argument—which we shall now continue.

Fundamental theorem

Equation (Template:EquationNote) is applicable where Template:Math is a scalar field,  Template:Mvar is a parameter measuring arc length along a curveTemplate:Math and Template:Math is the unit tangent vector toTemplate:Math in the direction of increasingTemplate:Mvar. Let Template:Mvar take the values Template:Math and Template:Math at the endpoints ofTemplate:Math where the position vector Template:Math takes the values Template:Math and Template:Math respectively. Then, integrating (Template:EquationNote) w.r.t.Template:Mvar from Template:Math to Template:Math and applying the fundamental theorem of calculus, we get Template:NumBlk This is our third integral theorem involving the gradient, and the best-known of the three: it is commonly called simply the gradient theorem, or the fundamental theorem of the gradient, or the fundamental theorem of line integrals; it generalizes the fundamental theorem of calculus to a curved path. If we write Template:Math  for  Template:Math (the change in the position vector), we get the theorem in the alternative form Template:NumBlk As the right-hand side of (Template:EquationNote) or (Template:EquationNote) obviously depends on the endpoints but not on the path in between, so does the integral on the left. This integral is commonly called the work integral ofTemplate:Math over the path—because if Template:Math is a force, the integral is the work done by the force over the path. So, in words, the gradient theorem says that the change in value of a scalar field from one point to another is the work integral of the gradient of that field field over any path from the one to the other.

Applying (Template:EquationNote) to a single element of the curve, we get Template:NumBlk which is reminiscent of  $y^{\prime }(x)dx=dy$ in elementary calculus. Alternatively, we could have obtained (Template:EquationNote) by multiplying both sides of (Template:EquationNote) byTemplate:Mvar, and then obtained (Template:EquationNote) by adding (Template:EquationNote) over all the elemental displacementsTemplate:Math on any path from Template:Math toTemplate:Math.

If we close the path by setting  Template:Math the gradient theorem reduces to Template:NumBlk where the integral is around any closed loop. Applying the Kelvin–Stokes theorem then gives Template:NumBlk where Template:Mvar is any surface spanning the loop and$\hat{\mathit{\nu }}$ is the unit normal toTemplate:Mvar.  As this applies to any loop spanned by any surface on which the integrand is defined,  Template:Math  must be zero wherever it is defined. This is a second proof (more conventional than the first) of theorem (Template:EquationNote).

Scalar potential: field with given gradient

Lemma:  If  Template:Math  in a simply connected regionTemplate:Mvar,  then  $\int 𝐪\cdot d𝐫$ over any path inTemplate:Mvar  depends only on the endpoints of the path.

Proof:  Suppose, on the contrary, that there are two paths Template:Math and Template:Math inTemplate:Mvar,  with a common starting point and a common finishing point, such that

$\underset{\mathrm{\Gamma }}{\int }𝐪\cdot d𝐫\ne \underset{\mathrm{\Lambda }}{\int }𝐪\cdot d𝐫.$

Let  Template:Math denote Template:Math traversed backwards. Then for every Template:Math on Template:Math  there is an equal and oppositeTemplate:Math on  Template:Math so that we have

$\underset{\mathrm{\Gamma }}{\int }𝐪\cdot d𝐫\ne -\underset{-\mathrm{\Lambda }}{\int }𝐪\cdot d𝐫,$

i.e.

$\underset{\mathrm{\Gamma }}{\int }𝐪\cdot d𝐫+\underset{-\mathrm{\Lambda }}{\int }𝐪\cdot d𝐫\ne 0,$

where the left-hand side is now a work integral ofTemplate:Math around a closed loop inTemplate:Mvar.  By the simple connectedness ofTemplate:Mvar,  this loop is spanned by some surfaceTemplate:Mvar inTemplate:Mvar.  So we can apply the Kelvin–Stokes theorem and conclude that the flux integral of  Template:Math  throughTemplate:Mvar  is non-zero, in which case  Template:Math  must be non-zero somewhere onTemplate:Mvar hence somewhere inTemplate:Mvar — contradicting the hypothesis of the lemma. ◼

Corollary:  If  Template:Math  in a simply connected regionTemplate:Mvar,  there exists a scalar field Template:Mvar such that  Template:Math  inTemplate:Mvar.

Proof:  We shall show that a suitable candidate is

$p(𝐫)=\underset{{𝐫}_0}{\overset{𝐫}{\int }}𝐪\cdot d\mathit{\rho },$

where Template:Math is the position vector of any fixed point inTemplate:Mvar,  and Template:Mvar is the position vector of a general point on the path of integration, which may be any path inTemplate:Mvar. First note that Template:Math is unambiguous because, by the preceding lemma, it is independent of the path for given Template:Math andTemplate:Math provided that the path is inTemplate:Mvar.  Now to find  Template:Math  let Template:Mvar be the arc length along the path from Template:Math toTemplate:Mvar, so that Template:Mvar ranges from 0 to (say)Template:Mvar  as Template:Mvar ranges from Template:Math toTemplate:Math; and let Template:Math be the unit vector tangential to the path atTemplate:Mvar, in the direction of increasingTemplate:Mvar.  Then  Template:Math so that the above equation becomes

$p(𝐫(s))=\underset{0}{\overset{s}{\int }}𝐪\cdot \widehat{𝐬}d\sigma .$

Differentiating w.r.t.Template:Mvar gives

${\partial }_{s}p=𝐪\cdot \widehat{𝐬},$

where Template:Math is evaluated at  Template:Mvar  and is therefore in the direction in which the path reachesTemplate:Math.  By the generality of the path, this can be any direction. So the last equation says that Template:Math is the vector whose (scalar) component in any direction is the derivative ofTemplate:Mvar w.r.t. arc length in that direction; that is, Template:Math as required. ◼

This is the promised converse of theorem (Template:EquationNote). But, given an irrotational vector field Template:Math we usually prefer to find a scalar field whose negative gradient isTemplate:Math;  that is, we usually prefer a scalar field $\phi$ such that  $𝐪=-\mathrm{\nabla }\phi$.  Such a field $\phi$ is called a scalar potential forTemplate:Math.  From the above expression for Template:Math a suitable candidate is Template:NumBlk

A scalar field has zero gradient if and only if it is uniform, so that adding a uniform field, but only a uniform field, to a given scalar field leaves its gradient unchanged. Thus the scalar potential is determined up to an arbitrary additive uniform field. This would be the case with or without the minus sign in front of the gradient. The reason for preferring the minus sign appears next.

Conservative fields

An irrotational vector field—or, equivalently, a field that is (plus or minus) the gradient of something—is described as conservative, because if the field is a force, it does zero work around a closed loop, and consequently conserves energy around the loop (at least if the field does not change during traversal of the loop).

If the only force acting on a particle is  Template:Math  then, by the gradient theorem, the work done on the particle over a path is the increase in Template:Mvar,  i.e. the decrease inTemplate:Mvar; and this work is the increase in the particle's kinetic energyTemplate:Mvar.  Hence, if we identify Template:Mvar with the potential energy, the total energy  Template:Mvar  is conserved. This interpretation of the scalar potential is possible only if the force is minus the gradient of the potential.

The minus sign is also used if the conservative vector field is an electric field (force per unit charge) or a gravitational acceleration (force per unit mass); the scalar potential is potential energy per unit charge, or potential energy per unit mass, respectively.

Some special fields

The 1/r scalar potential

For the potential energy field Template:NumBlk where Template:Mvar is the distance from the origin (and Template:Math), let us find the corresponding force  Template:Math.  The direction of  Template:Math  is that of the steepest increase ofTemplate:Mvar, which, by the spherical symmetry, can only be parallel or antiparallel to Template:Math (the unit vector pointing away from the origin). So

$\mathrm{\nabla }U=(\mathrm{\nabla }U\cdot \widehat{𝐫})\widehat{𝐫}={\partial }_{r}U\widehat{𝐫}=\frac{d}{dr}\left(\frac{1}{r}\right)\widehat{𝐫}=-\frac{1}{r^2}\widehat{𝐫},$

whence Template:NumBlk So the negative gradient of the Template:Math  scalar potential (Template:EquationNote) is the unit inverse-square radial vector field. Multiplying the numerator and denominator by Template:Mvar gives the alternative form

$𝐅=\frac{𝐫}{r^3},$

which is convenient if the center of the force is shifted from the origin to positionTemplate:Math: in that case we simply replace Template:Math by Template:Math and Template:Mvar by Template:Math so that the force becomes

$𝐅=\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }{|}^3}$

and the corresponding scalar potential becomes

$U=\frac{1}{|𝐫-{𝐫}^{\prime }|}.$

Inverse-square radial vector field

We derived the vector field (Template:EquationNote) as the negative gradient of the scalar potential (Template:EquationNote). Conversely, given the inverse-square radial vector field (Template:EquationNote), we could derive its scalar potential from (Template:EquationNote). At a general point on the path, let the position vector be  $\mathit{\rho }=\rho \hat{\mathit{\rho }}$ so that, by (Template:EquationNote),  $𝐅=\hat{\mathit{\rho }}/{\rho }^2$.  Then (Template:EquationNote) becomes

$\begin{array}{cc}U(𝐫)& =-\underset{{𝐫}_0}{\overset{𝐫}{\int }}𝐅\cdot d\mathit{\rho }\\ & =-\underset{{𝐫}_0}{\overset{𝐫}{\int }}\frac{1}{{\rho }^2}\hat{\mathit{\rho }}\cdot d\mathit{\rho }\\ & =-\underset{r_0}{\overset{r}{\int }}\frac{1}{{\rho }^2}d\rho =\frac{1}{\rho }{|}_{r_0}^r=\frac{1}{r}-\frac{1}{r_0},\end{array}$

so that, if we choose  Template:Math we recover (Template:EquationNote).

Because Template:Math given by (Template:EquationNote), has a scalar potential,  Template:Math  must be zero. This is independently obvious in that the spherical symmetry ofTemplate:Math seems to rule out any resemblance of rotation or shear—even at the origin, where Template:Math becomes infinite. On the last point, let us check whether  Template:Math  has a meaningful integral over a volume containing the origin. If the volume Template:Mvar  is enclosed by the surface Template:Mvar  whose outward unit normal is Template:Math, then, by theorem (Template:EquationNote),

$\underset{V}{\iiint }\mathrm{curl}𝐅dV=\underset{S}{\iint }\widehat{𝐧}\times 𝐅dS=\underset{S}{\iint }\widehat{𝐧}\times \frac{\widehat{𝐫}}{r^2}dS.$

If Template:Mvar contains the origin, then, because  Template:Math  is zero everywhere except at the origin, the volume Template:Mvar  can be replaced by any element ofTemplate:Mvar  containing the origin, whatever the shape of that element may be. If we choose that element to be a spherical ball centered on the origin, then Template:Math is parallel to Template:Math so that the cross-product in the integrand on the right is zero. Thus the volume integral on the left is not only meaningful, but is zero, even if the volume contains the point where the integrand is infinite. In this sense, the field Template:Math is so irrotational that its curl may be taken as zero even where the field itself is undefined!

The situation concerning the divergence ofTemplate:Math is more complicated. Again, let the volume Template:Mvar  be enclosed by the surface Template:Mvar whose outward unit normal is Template:Math.  By the divergence theorem (Template:EquationNote),

$\begin{array}{cc}\underset{V}{\iiint }\mathrm{div}𝐅dV=\underset{S}{\iint }\widehat{𝐧}\cdot 𝐅dS& =\underset{S}{\iint }\widehat{𝐧}\cdot \frac{\widehat{𝐫}}{r^2}dS\\ & =\underset{S}{\iint }\frac{\widehat{𝐫}\cdot \widehat{𝐧}dS}{r^2}\\ & =\underset{S}{\iint }d\mathrm{\Omega },\end{array}$

where Template:Math is the solid angle subtended at the origin by the surface element of areaTemplate:Mvar, and is positive if the outward unit normal Template:Math has a positive component away from the origin (Template:Math), and negative if Template:Math has a positive component toward the origin (Template:Math). If the volume enclosed by Template:Mvar does not include the origin, then for every positive contribution Template:Math there is a compensating negative contribution, so that the integral of  Template:Math  over the volume is zero. As this applies to every such volume,  Template:Math  must be zero everywhere except at the origin. If, on the contrary, the volume does include the origin, then the contributions Template:Math add up to the total solid angle subtended by the enclosing surface, which isTemplate:Math. In summary, Template:NumBlk where Template:Math the 3D unit delta function, is zero everywhere except at the origin, but has an integral of  Template:Math over any volume that includes the origin. For example, a unit point-mass at the origin has the density Template:Math, and a point-mass Template:Mvar at position Template:Math has the density  Template:Math. As the argument of  Template:Math  in (Template:EquationNote) is  Template:Math we also have Template:NumBlk If we shift the centers from the origin to Template:Math the last two results become Template:NumBlk and Template:NumBlk

Field with given divergence (and zero curl)

It follows from Coulomb's law that the electric field due to a point-charge Template:Mvar at the origin, in a vacuum, is

$𝐄=\frac{Q}{4\pi {ϵ}_0{r}^2}\widehat{𝐫},$

where Template:Math is a physical constant (called the vacuum permittivity or simply the electric constant). In a vacuum, the electric displacement field, denoted byTemplate:Math is Template:Math.  So it is convenient to multiply the above equation by Template:Math obtaining

$𝐃=\frac{Q}{4\pi }\frac{\widehat{𝐫}}{r^2}.$

This is a inverse-square radial vector field and therefore has zero curl.

Now suppose that, instead of a charge Template:Mvar at the origin, we have a static charge density Template:Math in a general elemental volume Template:Mvar  at positionTemplate:Math (the standard symbol for charge density being unfortunately the same as for mass density). Then the contribution from that element to the fieldTemplate:Math at positionTemplate:Math  is

$d𝐃(𝐫)=\frac{\rho ({𝐫}^{\prime })d{V}^{\prime }}{4\pi }\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }{|}^3}$

provided that, for each Template:Math the dimensions of each volume element are small compared with Template:Math. This contribution likewise has zero curl. The total field due to static charges is then the sum of the contributions: Template:NumBlk where the integral is over all space. And Template:Math has zero curl because all the contributions have zero curl.

Independently of the physical significance of  Template:Math we can take its divergence "term by term" (or "under the integral sign"), obtaining

$\begin{array}{cc}\mathrm{div}𝐃(𝐫)& =\iiint \frac{\rho ({𝐫}^{\prime })}{4\pi }\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }{|}^3}\right)d{V}^{\prime }\\ & =\iiint \frac{\rho ({𝐫}^{\prime })}{4\pi }4\pi \delta (𝐫-{𝐫}^{\prime })d{V}^{\prime }\left[𝖻𝗒𝖾𝗊\mathsf{.}\mathsf{(}\mathrm{𝟥}\mathrm{𝟥}𝖽\mathsf{)}\right]\\ & =\iiint \rho ({𝐫}^{\prime })\delta (𝐫-{𝐫}^{\prime })d{V}^{\prime }\\ & =\iiint \rho (𝐫)\delta (𝐫-{𝐫}^{\prime })d{V}^{\prime }\left[\begin{array}{c}𝗌𝗂𝗇𝖼𝖾\delta (𝐫-{𝐫}^{\prime })=0\\ 𝗎𝗇𝗅𝖾𝗌𝗌{𝐫}^{\prime }=𝐫\end{array}\right]\\ & =\rho (𝐫)\iiint \delta (𝐫-{𝐫}^{\prime })d{V}^{\prime }\\ & =\rho (𝐫)\iiint \delta ({𝐫}^{\prime }-𝐫)d{V}^{\prime },\end{array}$

where the last step is permitted because the volume integral of the delta function ofTemplate:Math is not changed by a "point reflection" (inversion) across Template:Math.  As the volume of integration (all space) includes the shifted origin of the delta function, the integral is simplyTemplate:Math so that Template:NumBlk where both sides are evaluated atTemplate:Math.

Mathematically, this result is an identity which applies if  Template:Math is given by (Template:EquationNote); substituting forTemplate:Math we can write the identity in full as Template:NumBlk where the integral is over all space, or at least all of the space in which Template:Mvar may be non-zero. Subject to the convergence of the integral, this shows that we can construct an irrotational vector field whose divergence is a given scalar fieldTemplate:Math. And of course, by theorem (Template:EquationNote), any curl can be added to that vector field without changing its divergence.

In electrostatics, (Template:EquationNote) is a generalization of Coulomb's law; and (Template:EquationNote), which follows from (Template:EquationNote), is Gauss's law expressed in differential form. If we integrate (Template:EquationNote) over a volume enclosed by a surfaceTemplate:Mvar (with outward unit normal Template:Math) and apply the divergence theorem on the left, we get the integral form of Gauss's law: Template:NumBlk where Template:Math is the total charge enclosed  byTemplate:Mvar.

Field with given Laplacian

In (Template:EquationNote), we can recognize the Template:Math-dependent factor  Template:Math  as  Template:Math  and take the gradient operator outside the integral, obtaining

$\rho (𝐫)\equiv \mathrm{d}\mathrm{i}\mathrm{v}(-\mathrm{\nabla }\iiint \frac{\rho ({𝐫}^{\prime })}{4\pi }\frac{1}{|𝐫-{𝐫}^{\prime }|}d{V}^{\prime }),$

i.e. Template:NumBlk where again the integral is over all space, or at least all of the space in which Template:Mvar may be non-zero. Subject to the convergence of the integral, this shows that we can construct a field whose Laplacian is a given field. More precisely, it shows that we can construct a scalar field whose Laplacian is a given scalar  fieldTemplate:Math. But, due to the linearity of the Laplacian, the same applies to any given linear combination of scalar fields, including any combination whose coefficients are uniform vectors, uniform matrices, or uniform tensors of any order; that is, the same applies to any field that we can express with a uniform basis.

Mathematically, (Template:EquationNote) is simply an identity. To find its significance in electrostatics, we can multiply it by  Template:Math obtaining Template:NumBlk which is also an identity. But the negative gradient of the expression after the integral sign is

$\frac{\rho ({𝐫}^{\prime })d{V}^{\prime }}{4\pi {ϵ}_0}\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }{|}^3},$

which is the contribution to the electric field at positionTemplate:Math due to a charge  Template:Math  at positionTemplate:Math in a vacuum. So the expression after the integral sign is the corresponding contribution to the electrostatic potential, and the whole integral is the whole electrostatic potential. Denoting this by $\phi ,$ we can rewrite (Template:EquationNote) as Template:NumBlk This is Poisson's equation in electrostatics, treating the medium as a vacuum (so that Template:Mvarmust be taken as the total charge density, including any contributions caused by the effect of the field on the medium). In a region in which  Template:Math  Poisson's equation (Template:EquationNote) reduces to Template:NumBlk which is Laplace's equation in electrostatics.

The wave equation

It is an empirical fact that a compressible fluid, such as air, carries waves of a mechanical nature: sound waves. In establishing the unambiguity of the gradient and the divergence, we have already derived equations dealing with the inertia and continuity (mass-conservation) of non-viscous fluids. So, by introducing a relation describing the compressibility, and eliminating variables, we should be able to get one equation (the "wave equation") in one scalar or vector field (the "wave function"), with recognizably "wavelike" solutions. And we should expect this equation to be analogous to equations describing other kinds of waves.

If we suppose, for simplicity, that the only force acting on an element of fluid is the pressure force, the applicable equation of motion is (Template:EquationNote). But, for reasons which will soon be apparent, let us call the pressureTemplate:Mvar, so that (Template:EquationNote) becomes

$\rho \frac{d𝐯}{dt}=-\mathrm{\nabla }P.$

Then at equilibrium we have

$0=-\mathrm{\nabla }{P}_0,$

where Template:Math is the equilibrium pressure. Subtracting this equation from the previous one and defining

$p=P-P_0,$

we get

$\rho \frac{d𝐯}{dt}=-\mathrm{\nabla }p,$

which looks like (Template:EquationNote), except that Template:Mvar is now the sound pressure (also called "acoustic pressure", or sometimes "excess pressure"), i.e. the pressure rise above equilibrium.

For the equation of continuity we can use (Template:EquationNote), which we repeat for convenience:

$\rho \mathrm{div}𝐯=-\frac{d\rho }{dt}.$

Eliminating Template:Math between the last two equations is fraught because Template:Mathis evaluated at a moving point in the former and at a fixed point in the latter; and introducing any relation between Template:Mvar and Template:Mvar is similarly fraught because Template:Mvar is evaluated at a fixed point and Template:Mvar at a moving point. The obvious remedy is to apply the advection rule (Template:EquationNote) to the last two equations, obtaining respectively

$\begin{array}{cc}\rho (\frac{\partial 𝐯}{\partial t}+𝐯\cdot \mathrm{\nabla }𝐯)& =-\mathrm{\nabla }p;\\ \rho \mathrm{div}𝐯& =-\frac{\partial \rho }{\partial t}-𝐯\cdot \mathrm{\nabla }\rho .\end{array}$

That gets all the variables evaluated at fixed points, at the cost of making the equations more complicated and more obviously non-linear. But the equations and be simplified and linearized by small-amplitude approximations. In the parentheses in the first equation, the first term is proportional to the amplitude of the vibrations while the second term is a product of two factors proportional to the amplitude, so that, for sufficiently small amplitudes, the second term is negligible. Similarly, in the second equation, for sufficiently small amplitudes and a homogeneous medium, we can neglect the second term on the right. Then, on the left side of each equation, we are left with a factor proportional to the amplitude, multiplied byTemplate:Mvar. But Template:Mvar is not proportional to the amplitude; only its deviation from the equilibrium density is so proportional. Hence, for small amplitudes,  Template:Mvarcan be replaced by the equilibrium density, which we shall callTemplate:Math which is independent of time and (in a homogeneous medium) independent of position. With these approximations, our equations of motion and continuity become

$\begin{array}{cc}{\rho }_0\dot{𝐯}& =-\mathrm{\nabla }p,\\ {\rho }_0\mathrm{div}𝐯& =-\dot{\rho },\end{array}$

where, for brevity, we use an overdot to denote partial differentiation w.r.t. time (i.e., at a fixed point, not a point moving with the fluid).

Now we can eliminate Template:Math. Taking divergences in the first equation, and differentiating the second partially w.r.t. time (which can be done inside the Template:Math operator, which represents a linear combination), we get

$\begin{array}{cc}{\rho }_0\mathrm{div}\dot{𝐯}& =-\mathrm{△}p,\\ {\rho }_0\mathrm{div}\dot{𝐯}& =-\ddot{\rho },\end{array}$

so that we can equate the right-hand sides, obtaining Template:NumBlk

Maintaining the small-amplitude assumption, we can now consider compressibility. For small compressions in a homogeneous medium, we may suppose that the pressure change Template:Mvar is some constant times the density changeTemplate:Mvar. It is readily verified that such a constant must have the dimension of velocity squared. So we can say  Template:Math where Template:Mvar is a constant with the units of velocity.Template:Efn Dividing by Template:Mvar gives  $\dot{p}=c^2\dot{\rho },$ whence Template:NumBlk Substituting from (Template:EquationNote) then gives the desired wave equation: Template:NumBlk

This is the 3D classical wave equation with the sound pressure Template:Mvar as the wave function. For a generic wave function Template:Mvar in a homogeneous isotropic medium, we would expect the equation to be Template:NumBlk which may be written more compactly as Template:NumBlk where Template:Math pronounced "wave" or "box",Template:Efn is called the D'Alembertian operator and is defined by Template:NumBlk in this paper, although other conventions exist.Template:Efn

In a static situation, the second term on the right of (Template:EquationNote) is zero. So one advantage of definition (Template:EquationNote), over any alternative definition that changes the sign or the scale factor, is that in the static case, the D'Alembertian is reduced to the Laplacian, making it especially obvious that in the static case, the wave equation is reduced to Laplace's equation [compare (Template:EquationNote) and (Template:EquationNote)]. Also notice that the D'Alembertian, being a linear combination of two linear operators, is itself linear.

Spherical waves

Having established that there are wavelike time-dependent fields described by equation (Template:EquationNote), in which the constant Template:Mvar has the units of velocity, we can now make an informed guess at an elementary solution of the equation. Consider the candidate Template:NumBlk where  Template:Math  is the position vector (so that Template:Mvar is distance from the origin),  Template:Mvar  is an arbitrary function (arbitrary except that it will need to be twice differentiable),  Template:Mvaris time, and Template:Mvaris a constant (and obviously Template:Mvaris not defined at the origin even if Template:Mvaris.)

If, at the origin, the function Template:Mvar  has a certain argument at time  Template:Math  then at any distanceTemplate:Mvar  from the origin, it has the same argument at time  Template:Math  which is  Template:Math later  than at the origin. Hence, if Template:Mvar  has a certain feature (e.g., a zero-crossing) at the origin, the time taken for that feature to reach any distanceTemplate:Mvar  isTemplate:Math  implying that the feature travels outward from the origin at speedTemplate:Mvar.  Another way to perceive this is to set the argument ofTemplate:Mvar  equal to a constant (corresponding to some feature of the function) and differentiate w.r.t.Template:Mvar obtaining  Template:Math  (the speed at which the feature recedes from the origin). Thus equation (Template:EquationNote) describes waves  radiating outward from the origin with speedTemplate:Mvar. Template:Efn

Equation (Template:EquationNote) further implies that there are surfaces over which the wave function Template:Mvar  is uniform—namely surfaces of constantTemplate:Mvar,  i.e. spheres centered on the origin. These are the wavefronts. So (Template:EquationNote) describes spherical waves.

Because the surface area of a sphere is proportional to the square of its radius, we should expect the radiated intensity (power per unit area) to satisfy an inverse-square law (if the medium is lossless—neither absorbing nor scattering the radiated power). That does not mean that the wave function itself should satisfy an inverse-square law. In a traveling wave in 3D space, there will be an "effort" variable (e.g., sound pressure) and a "flow" variable (e.g., fluid velocity), and the instantaneous intensity will be proportional to the product of the two. If the two are proportional to each other, the instantaneous intensity will be proportional to the square of one or the other. Hence if the instantaneous intensity falls off likeTemplate:Math the effort and flow variables—and the wave function, if it is proportional to one or the other—will fall off likeTemplate:Math. That suggests the attenuation factor Template:Math  in (Template:EquationNote).

But there are big if s in that argument. For all we know so far, the relation between effort and flow could involve a lag, so that the instantaneous product of the two could swing negative although it averages to something positive. And for all we know so far, the lag could vary withTemplate:Mvar, allowing at least one of the two (effort or flow) to depart from the Template:Math  law, even if their average product still falls off likeTemplate:Math. The Template:Math  factor in (Template:EquationNote) is therefore only an "informed guess". Notwithstanding these complications, we have also guessed that the form of the function Template:Mvar  (the waveform) does not change as Template:Mvar increases; we have not considered whether this behavior might depend on the medium, or the waveform, or the geometry of the wavefronts.

So let us carefully check whether (Template:EquationNote) satisfies (Template:EquationNote) or, equivalently, (Template:EquationNote).

As a first step, and as a useful inquiry in its own right, we find Template:Math from definition (Template:EquationNote), given that Template:Mvar is a function of Template:Mathonly. For the surface Template:Mvar  let us start with

• a cone (not a double cone) with its apex at the origin, subtending a small solid angle Template:Mvar at the origin,
• a sphere centered on the origin, with radius Template:Math, and
• a sphere centered on the origin, with radius Template:Mvar;

and let the volume element be the region inside the cone and between the spheres, so that its enclosing surface Template:Mvar  has three faces: a segment of the cone, a segment of the inner sphere with areaTemplate:Math and a segment of the outer sphere with areaTemplate:Math. By the symmetry ofTemplate:Mvar, the outward normal derivative Template:Mvar  is equal to zero on the conical face,  Template:Math on the outer spherical face, and  Template:Math on the inner spherical face. The volume of the element is  Template:Math. So, assembling the pieces of definition (Template:EquationNote), we get

$\begin{array}{cc}\mathrm{△}\psi & =\frac{1}{r^2\omega dr}((r+dr{)}^2\omega {\partial }_r\psi (r+dr,t)-r^2\omega {\partial }_r\psi (r,t))\\ & =\frac{1}{r^2}\frac{(r+dr{)}^2{\partial }_r\psi (r+dr,t)-r^2{\partial }_r\psi (r,t)}{dr}\\ & =\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{\partial }_r\psi (r,t)),\end{array}$

i.e. Template:NumBlk

Now we can verify our "informed guess". Differentiating (Template:EquationNote) twice w.r.t.Template:Mvar  by the chain rule gives Template:NumBlk where each prime Template:Math denotes differentiation of the function w.r.t. its own argument. Differentiating (Template:EquationNote) once w.r.t.Template:Mvar  by the product rule and chain rule, we get Template:NumBlk Proceeding as specified in (Template:EquationNote), we multiply this by Template:Math, differentiate again w.r.t.Template:Mvar (giving three terms, of which two cancel), and divide by Template:Math, obtaining Template:NumBlk Then if we substitute (Template:EquationNote) and (Template:EquationNote) into (Template:EquationNote), we obviously get  Template:Math satisfying (Template:EquationNote). So we have guessed correctly.

Having shown that the D'Alembertian ofTemplate:Mvar, as given by (Template:EquationNote), is zero everywhere except at the origin (where it is not defined), let us now find its integral over a volumeTemplate:Mvar (enclosed by a surfaceTemplate:Mvar) that includes the origin. From (Template:EquationNote),

$\begin{array}{cc}\underset{V}{\iiint }◻\psi dV& =\underset{V}{\iiint }\mathrm{△}\psi dV-\frac{1}{c^2}\underset{V}{\iiint }\frac{{\partial }^2\psi }{\partial t^2}dV\\ & =\underset{S}{\iint }{\partial }_n\psi dS-\frac{1}{c^2}\underset{V}{\iiint }\frac{{\partial }^2\psi }{\partial t^2}dV,\end{array}$

where the second equality follows from theorem (Template:EquationNote). Now because the integrand on the left is zero except at the origin, anyTemplate:Mvar containing the origin will give the same integral. So for convenience, let Template:Mvar be a spherical ball of radiusTemplate:Mvar centered on the origin. Then, by the spherical symmetry ofTemplate:Mvar integration overTemplate:Mvar reduces to multiplication byTemplate:Math and Template:Mvar is equivalent toTemplate:Mvar and Template:Mvar can be taken asTemplate:Math. With these substitutions we have

$\underset{V}{\iiint }◻\psi dV=4\pi R^2\frac{\partial \psi }{\partial r}{|}_{r=R}-\frac{1}{c^2}\underset{0}{\overset{R}{\int }}\frac{{\partial }^2\psi }{\partial t^2}2\pi rdr$

or, substituting from (Template:EquationNote) and (Template:EquationNote),

$\begin{array}{cc}\underset{V}{\iiint }◻\psi dV=& 4\pi R^2(-\frac{1}{cR}{f}^{\prime }(t-R/c)-\frac{1}{R^2}f(t-R/c\left)\right)\\ & -\frac{1}{c^2}\underset{0}{\overset{R}{\int }}\frac{1}{r}{f}^{″}(t-r/c)2\pi rdr\\ =& -\frac{4\pi R}{c}{f}^{\prime }(t-R/c)-4\pi f(t-R/c)\\ & -\frac{2\pi }{c^2}\underset{0}{\overset{R}{\int }}{f}^{″}(t-r/c)dr.\end{array}$

Again noting that any Template:Mvar containing the origin will give the same volume integral, we can let Template:Mvar approach zero, with the result that the right-hand side approaches Template:Math. This is the integral ofTemplate:Math over any volume containing the origin, for Template:Mvar given by (Template:EquationNote). Meanwhile Template:Math is zero everywhere except that the origin. In summary, Template:NumBlk Shifting the center of the spherical waves from the origin to positionTemplate:Math we get Template:NumBlk

We shall refer to the field given by (Template:EquationNote) as the wave function due to a monopole source with strength Template:Math at the origin. The D'Alembertian of this wave function is given by (Template:EquationNote). Hence the field whose D'Alembertian is given by (Template:EquationNote) is the wave function due to a monopole source with strength Template:Math at positionTemplate:Math. In each case, the D'Alembertian is zero everywhere except at the source.

Field with given D'Alembertian

Now suppose that, instead of a wave source with strength Template:Math at the general positionTemplate:Math we have at that position a wave-source density$w({𝐫}^{\prime },t)$ in an elemental volume Template:Mvar, whose contribution to the wave function Template:Mvar at positionTemplate:Math  is

$d\psi (𝐫,t)=\frac{1}{|𝐫-{𝐫}^{\prime }|}w({𝐫}^{\prime },t-\frac{|𝐫-{𝐫}^{\prime }|}{c})d{V}^{\prime },$

where for each Template:Math the dimensions of each volume element are small compared with Template:Math. Then the total wave function is the sum of the contributions: Template:NumBlk where the integral is over all space.

Independently of the physical significance ofTemplate:Math we can take its D'Alembertian "under the integral sign" by rule (Template:EquationNote), obtaining

$\begin{array}{cc}◻\psi (𝐫,t)& =\iiint (-4\pi w({𝐫}^{\prime },t)\delta (𝐫-{𝐫}^{\prime }))d{V}^{\prime }\\ & =\iiint (-4\pi w(𝐫,t)\delta (𝐫-{𝐫}^{\prime }))d{V}^{\prime }\\ & =-4\pi w(𝐫,t)\iiint \delta (𝐫-{𝐫}^{\prime })d{V}^{\prime }\\ & =-4\pi w(𝐫,t)\iiint \delta ({𝐫}^{\prime }-𝐫)d{V}^{\prime };\end{array}$

that is, Template:NumBlk

Mathematically, equation (Template:EquationNote) is an identity which applies if Template:Math is given by (Template:EquationNote). Substituting from (Template:EquationNote) and solving for$w,$ we can write the identity in full as Template:NumBlk where the integral is over all space, or at least all of the space in which$w$ may be non-zero. Subject to the convergence of the integral, this shows that we can construct a wave function with a given D'Alembertian.

Physically, equation (Template:EquationNote) gives the D'Alembertian of the wave function for a source density$w$. It is the inhomogeneous wave equation, which applies in the presence of an arbitrary source density—in contrast to the homogeneous wave equation (Template:EquationNote), which applies in a region where the source density is zero. In this context the word homogeneous or inhomogeneous describes the equation, not the medium (which has been assumed homogeneous and isotropic).

In a static situation, in which the D'Alembertian is reduced to the Laplacian, the inhomogeneous wave equation (Template:EquationNote) is reduced to the form of Poisson's equation (Template:EquationNote). As written, equation (Template:EquationNote) is Poisson's equation in electrostatics; it applies to the charge densityTemplate:Math, for which the scalar potential [in (Template:EquationNote)] is

$\phi (𝐫)=\frac{1}{4\pi {ϵ}_0}\iiint \frac{1}{|𝐫-{𝐫}^{\prime }|}\rho ({𝐫}^{\prime })d{V}^{\prime }.$

In electrodynamics, which takes time-dependence into account, the scalar potential due to the charge densityTemplate:Math is

$\phi (𝐫,t)=\frac{1}{4\pi {ϵ}_0}\iiint \frac{1}{|𝐫-{𝐫}^{\prime }|}\rho ({𝐫}^{\prime },t-\frac{|𝐫-{𝐫}^{\prime }|}{c})d{V}^{\prime },$

where the wave speed Template:Mvar is the speed of light; this is the same as in the static case except for the delay Template:Math indicating that the influence of the change density atTemplate:Math travels outward from that point at the speed of light. In the dynamic case, by rule (Template:EquationNote), the D'Alembertian of the scalar potential is

$◻\phi =-\frac{\rho (𝐫,t)}{{ϵ}_0}.$

This result is the inhomogeneous wave equation in the scalar potential—the equation which, in the electrostatic case, reduces to Poisson's equation (Template:EquationNote).

In electrodynamics, however, the electric field  Template:Math is not simply$-\mathrm{\nabla }\phi ,$ but$-\mathrm{\nabla }\phi -\frac{\partial 𝐀}{\partial t},$ where Template:Math is the magnetic vector potential, whose defining property is that its curl is the magnetic flux density:

$𝐁=\mathrm{curl}𝐀.$

By identity (Template:EquationNote), this property implies

$\mathrm{div}𝐁=0,$

which is Gauss's law for magnetism. We have noted in passing—but not yet proven—that (Template:EquationNote) has a converse, whereby the solenoidality ofTemplate:Math implies the existence of the vector potentialTemplate:Math.  Precedents suggest we might be able to prove this by finding a vector field whose curl is a delta function—perhaps through new identities relating it to a field whose divergence is a delta function—and using it to construct a vector field with a given curl. In fact we shall prove our "converse" differently, but we shall still need some new identities for the purpose. And to obtain those identities (among others), we must take the detour that we have made a virtue of not taking until now…

Cartesian coordinates

Indicial notation; implicit summation

Considering that a scalar field is a function of three coordinates, while a vector field has three components each of which is a function of three coordinates, we can readily imagine that coordinate-based derivations of vector-analytic identities are likely to be excruciatingly repetitive—unless perhaps we choose a notation that concisely specifies the repetition. So, instead of writing the Cartesian coordinates as Template:Math  we shall usually write them as Template:Mvar  where  Template:Math  respectively;  and instead of writing the unit vectors in the directions of the respective axes as Template:Math  we shall usually write them as Template:Math.  And for partial differentiation w.r.t.Template:Math instead of writing Template:Mvar or even Template:Math we shall write Template:Mvar.

Now comes a stroke of genius for which we are indebted to Einstein—although he used it in a more sophisticated context!  Instead of writing the position vector as

$𝐫=x_1{𝐞}_1+x_2{𝐞}_2+x_3{𝐞}_3$

or even as

Template:Big

we shall write it simply as

Template:Big

where it is understood  that we sum over the repeated index. More generally, we shall write the vector field Template:Math as

Template:Big

with implicit summation, and the vector field Template:Math as

Template:Big

with implicit summation, and so on. (By that nomenclature, the position vector in Cartesian coordinates should be, and often is, called Template:Math; but we called it Template:Math because we wanted to call its magnitude Template:Mvar, for radius.)

Implicit summation not only avoids writing the Template:Big symbol and specifying the index of summation, but also allows a summation over two repeated indices, say Template:Mvar and Template:Mvar, to be considered as summed first over Template:Mvar and then over Template:Mvar or vice versa, removing the need for an explicit regrouping of terms. Of course, if we hide messy details behind a notation, we need to make sure that it handles those details correctly. In particular, when we perform an operation on an implicit sum, we implicitly perform it term-by-term, and must therefore make sure that the operation is valid when interpreted that way.

Formulation of operators

Gradient:  Putting  Template:Mvar  in (Template:EquationNote), we find that the scalar component ofTemplate:Math in the direction of each Template:Math  isTemplate:Mvar.  To obtain the vector component in that direction, we multiply by Template:Math.  Assembling the components, we have (with implicit summation) Template:NumBlk or, in operational terms, Template:NumBlk or, in traditional longhand notation, Template:NumBlk It is also worth noting, from (Template:EquationNote), that the squared magnitude ofTemplate:Mathis Template:NumBlk where we write  Template:Mvar  rather than Template:Math  to ensure that implicit summation applies!

As reported by Tai (1994), there are unfortunately some textbooks in which the del operator is defined as

Template:BigTemplate:Big

—which, on its face, is not an operator at all, but a self-contained expression whose value is the zero vector (because it is a sum of derivatives of constant vectors). Among the offenders is Erwin Kreyszig, who, in the 6th edition of his bestselling Advanced Engineering Mathematics (1988, p. 486), misdefines the del operator thus and then rewrites the gradient ofTemplate:Mvar  as Template:Math apparently imagining that the differentiation operators look through the constant vectors rather than at  them. Six pages later, he defines the divergence in Cartesian coordinates (which we shall do shortly) and then immediately informs us that "Another common notation for the divergence ofTemplate:Math is Template:Math," where Template:Math is defined as before, but the resulting Template:Math is apparently not identically zero! These errors persist in the 10th edition (2011, pp. 396, 402–3). Tai finds similar howlers in mathematics texts by Wilfred Kaplan, Ladis D. Kovach, and Merle C. Potter, and in electromagnetics texts by William H. Hayt and Martin A. Plonus.  Knudsen & Katz, in Fluid Dynamics and Heat Transfer (1958), avoid the misdefinition ofTemplate:Math but implicitly define the divergence ofTemplate:Math as Template:Math  (which, as we have seen, is actually an operator), and then somehow reduce it to the correct expression forTemplate:Math.   But I digress.

Curl and divergence:  Expressing the operand of the curl in components, and noting that the unit vectors are uniform, we can apply (Template:EquationNote):

Template:Big

If we sum over Template:Mvar first, this is Template:NumBlk or, in operational terms, Template:NumBlk or, in traditional longhand,

Template:Big

For the divergence we proceed as for the curl except that, instead of (Template:EquationNote), we use (Template:EquationNote):

Template:Big

that is, Template:NumBlk or, in operational terms, Template:NumBlk or, in traditional longhand,

Template:Big

It follows from (Template:EquationNote) and (Template:EquationNote), if it was not already obvious, that a uniform vector field has zero curl and zero divergence.

Although the above expressions for the divergence and curl will surprise many modern readers, they match the initial definitions of the divergence and curl given by the founder of vector analysis as we know it, J. Willard Gibbs (1881, § 54). Gibbs even uses the Template:Math  and Template:Math  notations on the left sides of the defining equations, and only after  the equations (albeit immediately after) does he announce that  "Template:Math is called the divergence ofTemplate:Mvar  and Template:Math  its curl." (He uses Greek letters for vectors.) Our notation and Cartesian expression for the gradient (Template:EquationNote) also match Gibbs (1881, § 52). Hence, using the Gibbs notations, we can merge definitions (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) into the general Cartesian formula Template:NumBlk (with implicit summation), where the Template:Math operator may be a null (for the gradient), a cross (for the curl), or a dot (for the divergence).

Gibbs does not offer any justification for the Template:Math  and Template:Math  notations, but nor is it difficult to find such a justification based on his definitions. AsTemplate:Math is a uniform vector, we can rewrite (Template:EquationNote) rigorously as Template:NumBlk and thence operationally as Template:NumBlk or, recalling (Template:EquationNote),

$\mathrm{curl}𝐪=\mathrm{\nabla }\times 𝐪,$

which can be evaluated in the usual manner as

$\mathrm{curl}𝐪=\left|\begin{array}{ccc}𝐢& {\partial }_x& q_x\\ 𝐣& {\partial }_y& q_y\\ 𝐤& {\partial }_z& q_z\end{array}\right|,$

where Template:Mvar is the Template:Mvar component ofTemplate:Math, etc. This indeed is how one evaluates the curl of a given field in Cartesian coordinates, although we shall find (Template:EquationNote) more convenient for deriving identities. Similarly, we can rewrite (Template:EquationNote) rigorously as Template:NumBlk and thence operationally as Template:NumBlk or, recalling (Template:EquationNote),

$\mathrm{div}𝐪=\mathrm{\nabla }\cdot 𝐪.$

For evaluating the divergence of a given field, however, we simplify (Template:EquationNote) to

Template:Big

or, in traditional longhand,

Template:Big

although we shall find (Template:EquationNote) more convenient for deriving identities. But the longhand form makes it especially obvious that ifTemplate:Math is the position vector, Template:NumBlk

Notice that we can get from (Template:EquationNote) back to (Template:EquationNote) by permuting the Template:Mvar with the dot, and from (Template:EquationNote) back to (Template:EquationNote) by permuting the Template:Mvar with the cross, as if the differentiation operator could, as it were, look through the dot or the cross—or, as Gibbs's student Edwin B. Wilson puts it, "pass by" the dot and the cross, yielding Gibbs's original definitions. Hence Wilson considers it helpful to regard Gibbs's Template:Math  and Template:Math  notations as "the (formal) scalar product and the (formal) vector product ofTemplate:Math into" the operand, or "the symbolic scalar and vector products ofTemplate:Math into" the operand, and to regard Template:Math as a "symbolic vector" (not to be confused with Tai's symbolic vector$\mathrm{\nabla }{}^{{}_{-}}$).

Tai (1994, 1995) rejects Wilson's argument together with the entire tradition of treating Template:Math  and Template:Math  as compound operators. Of formal products, Tai says that the concept "has had a tremendously detrimental effect upon the learning of vector analysis"; he calls such a product a "meaningless assembly". Of the "pass by" step, he complains that "standard books on mathematical analysis do not have such a theorem."

I submit, however, that the intermediate steps (Template:EquationNote) and (Template:EquationNote), after which we take the constant multiplier outside the operator (eqs. Template:EquationNote & Template:EquationNote), support Wilson's "pass by" argument. In any event the reader may write out the sums on the right-hand sides of (Template:EquationNote) and (Template:EquationNote) and verify that they agree with the formal products Template:Math  and Template:Math  respectively—and may notice that in the evaluation of each formal product, the cross or dot may be eliminated, leaving nothing to "pass by". I further submit that the great generality of our derivation of equations (Template:EquationNote), above, compels us to treat the Template:Math  and Template:Math  notations as more than mere notations. But the kicker is that Tai himself, having found the form of the del operator in general coordinates (1995, p. 64, eq. 9.33), derives original corresponding forms of the Template:Math and Template:Math operators (his eqs. 9.35 & 9.40) which, upon reversal of the forbidden "pass by", become del-dot and del-cross! Indeed his three equations, just cited, are reminiscent of our (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) respectively. That being said, I shall find some points of agreement with Tai, and some reasons to criticize Wilson.

Laplacian:  If Template:Mvar is a scalar field, then

Template:Big

that is, Template:NumBlk where we write  Template:Mvar  rather than Template:Math  in order to maintain implicit summation. In traditional longhand, (Template:EquationNote) becomes

$\mathrm{△}\psi =\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}+\frac{{\partial }^2\psi }{\partial z^2}$

or, in operational terms,

$\mathrm{△}=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$

or, by comparison with (Template:EquationNote),

$\mathrm{△}=\mathrm{\nabla }\cdot \mathrm{\nabla }$

—as expected.

By the linearity of the Laplacian, the same applies if Template:Mvar is any field expressible in terms of a uniform basis. For example, if Template:Mvar is a vector field given by  Template:Math  (with implicit summation), then

Template:Big

where the third line follows from (Template:EquationNote) as applied to a scalar field. Thus (Template:EquationNote) is quite general.

After listing theorems (Template:EquationNote) to (Template:EquationNote) above, we gave reasons for describing Template:Math Template:Math and Template:Math as differential operators, and Template:Math as a 2nd-order differential operator—the implication being that the others are only 1st-order. We now have the promised "additional reason" for these descriptions: when expressed in Cartesian coordinates, the Template:Math operator involves second derivatives, while the others involve (only) first derivatives. In the meantime we have acquired the Template:Math operator, which is also 1st-order, as we shall now confirm.

Advection, directional derivative, etc.:  If Template:Mvar is a scalar field, then

Template:Big

In this double summation, the only non-zero terms are those for which  Template:Mvar,  in which case  Template:Math.  So we have Template:NumBlk or, in operational terms, Template:NumBlk or, in traditional longhand,

Template:Big

which indeed is the "formal" or "symbolic" dot-product of  Template:Math andTemplate:Math.  By the linearity of the directional derivative in (Template:EquationNote), the same result applies if Template:Mvar is a vector field or any field expressible in terms of a uniform basis. In particular, ifTemplate:Math is the position vector, we have

Template:Big

i.e., Template:NumBlk —which is also deducible from (Template:EquationNote).

For convenience in the following discussion, we shall refer to the scaled-directional-derivative operator Template:Math as an "advection" operator although, physically, it represents advection only if Template:Math is the material velocity.

Identities without pain

In deriving the Cartesian expressions for the gradient, curl, divergence, Laplacian, and advection operators, we used the preceding identities (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) respectively, the last being a definition generalizing (Template:EquationNote). Thus we could have derived the Cartesian expressions quite early in the exposition, although we did not find that option convenient. The other vector-analytic identities that we have previously mentioned are:

• (Template:EquationNote), which showed the unambiguity of the curl;
• (Template:EquationNote), which has a question mark after it;
• (Template:EquationNote), a product rule for the divergence, which is yet to be proven as a general identity;
• (Template:EquationNote) and (Template:EquationNote), concerning "curl grad" and "div curl"; and
• the identities showing that we can construct a field with a given divergence (Template:EquationNote), Laplacian (Template:EquationNote), or D'Alembertian (Template:EquationNote).

The above list exposes the following shortcomings:

• we have not yet investigated "grad div" and "curl curl";
• we have only one product rule —the unverified identity (Template:EquationNote)—in which both factors are spatially variable fields; this needs to be verified and identities (Template:EquationNote) and (Template:EquationNote) need to be generalized;
• our collection of product rules does not yet include the curl of a cross-product, or the gradient of a dot-product or of a product of scalars, or the advection of a product; and
• we do not yet have any chain rules involving Template:Math Template:Math or Template:Math.

With the aid of the Cartesian forms of the various operators, we may now fill these gaps.

The "grad div" and "curl curl" operators turn out to be related:

Template:Big

whence expanding the vector triple product gives

Template:Big

In the first term on the right, we can switch the order of partial differentiation; and in the second term—which, like the first, is a double summation—the only non-zero contributions are those for which  Template:Mvar  and  Template:Math.  So we have

Template:Big

that is, Template:NumBlk This result may be memorized as "curl curl is grad div minus del squared " and written as Template:NumBlk which looks like the expansion of a vector triple product; and the key step in the above derivation, based on the Gibbs definitions of the operators, really is  the expansion of a vector triple product.

We now turn to product rules in which neither factor is assumed uniform.

The curl of a cross-product is

Template:Big

i.e., Template:NumBlk

The divergence of a cross-product, as we might expect, is simpler:

Template:Big

i.e., Template:NumBlk In particular, in electromagnetics,  Template:Math ;  this is the identity on which Poynting's theorem is based. But if  Template:Math in (Template:EquationNote) is uniform, then (Template:EquationNote) reduces to (Template:EquationNote).

The gradient of a dot-product, by comparison, is surprisingly messy:

Template:Big

Now the first term on the right can be recognized as  Template:Math;  that is,  Template:Math;  that is,  Template:Midsize.  Similarly, the second term is  Template:Midsize.  Thus we have Template:NumBlk For uniform  Template:Math the first and third terms on the right vanish, and we can solve for the first term on the right, obtaining

$𝐛\times \mathrm{curl}𝐚=\mathrm{\nabla }𝐛\mathbf{\cdot }𝐚-𝐛\cdot \mathrm{\nabla }𝐚$[ for uniform Template:Math] ,

so that we can now drop the question mark after (Template:EquationNote). If we write the curl operator as  Template:Math  the last equation [or (Template:EquationNote)]  looks like the expansion of a vector triple product; but the identity is valid only for uniformTemplate:Math.

The gradient of a product of scalars, unlike that of a dot-product, is as simple as the product rule for ordinary differentiation:

Template:Big

that is, Template:NumBlk

The advection of a product is equally simple, regardless of the type of product, except that the order of a cross-product matters. Let Template:Mvar and Template:Mvar be scalar or vector fields, and let Template:Math denote any meaningful product of the two. Then, by (Template:EquationNote),

Template:Big

that is, Template:NumBlk The Template:Math operator is a scalar operator in the sense that it maps the operand field to a field of the same order—a scalar field to a scalar field, a vector field to a vector field, a matrix field to a matrix field, etc.— as if  it were multiplication by a scalar or differentiation w.r.t. a scalar; and indeed a differentiation w.r.t. path length appears in the coordinate-free definition (Template:EquationNote) of the operator. Moreover, we did not need coordinates to obtain rule (Template:EquationNote); as the reader may verify, the same rule can be obtained directly from the definition (Template:EquationNote) in a similar manner. From these points of view, the simplicity of the rule is unsurprising.

The curl of the product of a scalar and a vector is

Template:Big

that is, Template:NumBlk For uniform Template:Math this reduces to (Template:EquationNote), which was used to derive the Cartesian form of the curl (Template:EquationNote).

For the divergence of the product of a scalar and a vector, we proceed likewise except that we use a dot instead of a cross. The result is Template:NumBlk which has the same form as (Template:EquationNote), delivering the promised confirmation that (Template:EquationNote) is an identity. For uniform Template:Math  (Template:EquationNote) reduces to (Template:EquationNote), which was used to derive the Cartesian form of the divergence (Template:EquationNote).

That exhausts the first-order product rules. For curiosity's sake, we shall also derive one second-order rule.

The Laplacian of the product of a scalar field and a generic field, by (Template:EquationNote), is

Template:Big

In the middle term, by (Template:EquationNote), Template:Mvar  is the Template:Mvarth component ofTemplate:Math  so that, by (Template:EquationNote),  Template:Mvar  is the Template:Math operator for  Template:Math.  So we have Template:NumBlk The argument assumes a scalarTemplate:Mvar but is indifferent to whether Template:Mvar is a scalar or a vector or a higher-order tensor.

Finally we turn to chain rules — especially the simple cases of the gradient, curl, divergence, advection, and Laplacian of a function of a scalar fieldTemplate:Mvar. As usual, let Template:Mvar denote a scalar field, Template:Math a vector field, and Template:Mvar a generic field.

Gradient ⧸ curl ⧸ divergence of a function of a scalar:  By the general Cartesian formula (Template:EquationNote) and the chain rule forTemplate:Math

Template:Big

i.e., by (Template:EquationNote), Template:NumBlk In particular, if  Template:Math is a null, Template:NumBlk and if  Template:Math is a cross, Template:NumBlk and if  Template:Math is a dot, Template:NumBlk

Advection of a function of a scalar:

$\begin{array}{cc}𝐪\cdot \mathrm{\nabla }(\psi (u))& =q_i{\partial }_i(\psi (u))\\ & =q_i{\psi }^{\prime }(u){\partial }_{i}u\\ & =q_i{\partial }_{i}u{\psi }^{\prime }(u);\end{array}$

i.e., Template:NumBlk This fits into the pattern set by (Template:EquationNote) in that the gradient operator in (Template:EquationNote) is replaced by an advection operator.

Of the last four results, only (Template:EquationNote) is dependent on the order of the Template:Math product; the others could equally well be written Template:NumBlk

The Laplacian of a function of a scalar departs from the above pattern.

Template:Big

where the last line follows from the product rule for Template:Mvar  and, in the second term, the chain rule forTemplate:Mvar.  In that second term, the implicit sum  Template:Mvar  can be recognized as  Template:Math  by (Template:EquationNote). So we have Template:NumBlk

Multivariate chain rule:  The foregoing chain rules involve one intermediate function of one scalar variable. It will be useful to have an elementary chain rule that can handle more than one of each. Let Template:Math be a smooth scalar field, and let Template:Math in turn be a smooth function of several variables, one of which, sayTemplate:Mvar, is allowed to vary while the others are held constant, so that Template:Math changes by Template:Math when Template:Mvar changes by Template:Mvar. Then dividing (Template:EquationNote) by Template:Mvar  gives

${\partial }_{t}p=\mathrm{\nabla }p\cdot {\partial }_t𝐫$

or, in indicial Cartesian coordinates with implicit summation,

Template:Big

or, in traditional longhand,

Template:Big

This is the desired multivariate chain rule for a scalar function of three intermediate real variables. The assumption that these variables are Cartesian coordinates is not a loss of generality, because any three real quantities can be suitably scaled and represented by perpendicular axes, so that any scalar function of them becomes a function of position, to which (Template:EquationNote) applies; and then the scaling can be reversed without changing the products in the last equation. Moreover, by the linearity ofTemplate:Mvar, the scalar field Template:Mvar may be replaced by any field expressible in terms of a uniform basis. For example, for a vector fieldTemplate:Math,

Template:Big

where the third line is obtained by applying the multivariate chain rule for a scalar field. Thus, for a generic field Template:Mvar, Template:NumBlk

Gradient ⧸ curl ⧸ divergence of a function of a scaled position vector:  We end this subsection by deriving a lemma for use in the next subsection. IfTemplate:Mvar is a uniform scalar multiplier and Template:Math is the position vector,

Template:Big

where the third expression is obtained by from the second by multiplying each denominator (change inTemplate:Mvar) byTemplate:Mvar  and compensating. But now we have Template:NumBlk where the vertical bar and subscript indicate that the gradient, curl, or divergence is evaluated atTemplate:Math. We shall be interested in the curl (for which Template:Math is a cross).

Field with given curl

Consider the vector field Template:NumBlk where Template:Math is a solenoidal  vector field and Template:Mathis the position vector. By identity (Template:EquationNote),

$\mathrm{curl}𝐯=𝐪\mathrm{div}𝐫-𝐫\mathrm{div}𝐪+𝐫\cdot \mathrm{\nabla }𝐪-𝐪\cdot \mathrm{\nabla }𝐫$

where, by hypothesis, Template:Math  is zero. Applying identities (Template:EquationNote) and (Template:EquationNote) then yields

$\begin{array}{cc}\mathrm{curl}𝐯& =3𝐪+𝐫\cdot \mathrm{\nabla }𝐪-𝐪\\ & =2𝐪+𝐫\cdot \mathrm{\nabla }𝐪.\end{array}$

Animation of a rigid-body-like velocity field, whose curl is twice the angular velocity.

In the special case in which Template:Math is the angular velocity Template:Math of a rigid body about an axis through the origin,  Template:Mathis the velocity field (Template:Math) and Template:Math is uniform, so that the last result reduces to  Template:Math; that is, the vorticity is twice the angular velocity. As the vorticity in this case is uniform and therefore independent of position relative to the axis, it does not change if the axis is shifted, provided that the angular velocity has the same magnitude and direction. And because a uniform velocity field has zero curl, the vorticity is also unchanged if a translational motion is superposed on the rotation. This is the most direct connection that we have seen between curl and rotation. But again I digress.

Returning to the more general case in which Template:Mathis not necessarily uniform, but merely solenoidal, we have

$\mathrm{curl}𝐯(𝐫)=2𝐪(𝐫)+𝐫\cdot \mathrm{\nabla }𝐪(𝐫),$

to which we can apply our lemma (Template:EquationNote) with a uniform real factor Template:Mvar, obtaining

$\mathrm{curl}𝐯(t𝐫)=2t𝐪(t𝐫)+t𝐫\cdot \mathrm{\nabla }𝐪(t𝐫).$

On the left we can recall (Template:EquationNote); and on the right we can apply (Template:EquationNote), noting that the magnitude ofTemplate:Math isTemplate:Mvar, which measures distance in the direction ofTemplate:Math. Thus we obtain

$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}(𝐪(t𝐫)\times t𝐫)=2t𝐪(t𝐫)+tr{\partial }_r𝐪(t𝐫).$

Now if the direction ofTemplate:Math is held constant,  Template:Math is a function ofTemplate:Mvar; and in general  Template:Math.  So we have

$\begin{array}{cc}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}(𝐪(t𝐫)\times t𝐫)& =2t𝐪(t𝐫)+t^2{\partial }_t𝐪(t𝐫)\\ & ={\partial }_t(t^2𝐪(t𝐫)).\end{array}$

Integrating w.r.t. Template:Mvar  from 0 to 1 gives

$\underset{0}{\overset{1}{\int }}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}(𝐪(t𝐫)\times t𝐫)dt=(t^2𝐪(t𝐫)){|}_0^1=𝐪(𝐫);$

that is, Template:NumBlk Thus for any solenoidal vector fieldTemplate:Math  we can construct a vector potential—that is, a field whose curl isTemplate:Math; such a field is given by the integral on the right. This is the long-promised proof of the "converse" of identity (Template:EquationNote). Of course the vector potential is not unique, because any conservative field—but only a conservative field—can be added to it without changing its curl. Hence the existence of one vector potential implies the existence of infinitely many. The above integral gives us one.

The proof of (Template:EquationNote) assumes that Template:Mathis solenoidal not only at positionTemplate:Math but also atTemplate:Math  where  Template:Math, i.e. at every point on the line-segment from the origin toTemplate:Math.  A star-shaped region is one that contains an pointTemplate:Mvar  such that for every pointTemplate:Mvar in the region, the line-segment Template:Mvar is entirely contained in the region. We may choose any such Template:Mvar  as the origin in the proof of (Template:EquationNote). So the proof tells us that if a vector field is solenoidal within a star-shaped region, it has a vector potential in that region. As a special case, a vector field that is solenoidal everywhere has a vector potential everywhere.

Notes on the curl of the curl

Identity (Template:EquationNote), namely

$\mathrm{curl}\mathrm{curl}𝐪\equiv \mathrm{\nabla }\mathrm{div}𝐪-\mathrm{△}𝐪$

("curl curl is grad div minus del squared"), has at least three implications worth noting here.

First, it can be rearranged as Template:NumBlk ("del squared is grad div minus curl curl"). This would serve as a coordinate-free definition of the Laplacian of a vector, if we did not already have one. But we do: we started with a coordinate-free definition (Template:EquationNote) for a generic field, established its unambiguity via (Template:EquationNote), and found its Cartesian form (Template:EquationNote), which we used in the derivation of (Template:EquationNote). Wherever we start, we may properly assert by way of contrast that the Laplacian of a vector  is given by (Template:EquationNote), whereas the Laplacian of a scalar  is given by the divergence of the gradient. But we should not conclude, as Moon & Spencer do, that representing the scalar and vector Laplacians by the same symbol is "poor practice… since the two are basically quite different", because in fact the two have a common definition which is succinct, unambiguous, and coordinate-free: the Laplacian (of anything) is the closed-surface integral of the outward normal derivative, per unit volume.Template:Efn

Second, by reason of identity (Template:EquationNote) and the remarks thereunder, a given vector fieldTemplate:Math can be written

$𝐯(𝐫)\equiv \mathrm{△}(-\iiint \frac{𝐯({𝐫}^{\prime })}{4\pi }\frac{1}{|𝐫-{𝐫}^{\prime }|}d{V}^{\prime }),$

where the integral is over all space, or at least all of the space in which Template:Math may be non-zero. So, subject to the convergence of the integral, there exists a vector fieldTemplate:Math such that

$𝐯=\mathrm{△}𝐪;$

that is, by (Template:EquationNote), there existsTemplate:Math such that

$𝐯=\mathrm{\nabla }\mathrm{div}𝐪-\mathrm{curl}\mathrm{curl}𝐪,$

which implies the existence of a scalar field, say$\phi ,$ and a vector field, sayTemplate:Math, such that

$𝐯=-\mathrm{\nabla }\phi +\mathrm{curl}\mathrm{\Psi }$

(namely  Template:Midsize and  Template:Math). In short, subject to the convergence of the said integral,

• a given vector field can be resolved into [minus] a gradient plus a curl.

Such a resolution is called a Helmholtz decomposition, and the proposition that it exists is the Helmholtz decomposition theorem. Of course the gradient is irrotational and the curl is solenoidal so that, subject to the same convergence,

• a given vector field can be resolved into an irrotational field plus a solenoidal field.

This is a second statement of the theorem, and follows from the first. And the first follows from the second because an irrotational field has a scalar potential by (Template:EquationNote) and a solenoidal field has a vector potential by (Template:EquationNote).

Third, if Template:Math is solenoidal, the term  Template:Math  in (Template:EquationNote) or (Template:EquationNote) vanishes. Hence for a solenoidal field, the curl of the curl is minus the Laplacian. For example, in the dynamic case, in a vacuum, the Maxwell–Ampère law says that  Template:Math.  Multiplying this by the physical constant Template:Math (called the vacuum permeability or simply the magnetic constant) gives  Template:Math  whence

$\mathrm{curl}\mathrm{curl}𝐁={\mu }_0{ϵ}_0\mathrm{curl}\dot{𝐄}.$

But, by Gauss's law for magnetism, Template:Math is solenoidal so that, by (Template:EquationNote), the left-hand side of the above is  Template:Math.  And by Faraday's law,  Template:Math  so that  Template:Midsize.  Making these substitutions, we get  Template:Midsize i.e.

$\ddot{𝐁}=\frac{1}{{\mu }_0{ϵ}_0}\mathrm{△}𝐁.$

By comparison with (Template:EquationNote), this is the wave equation with

$c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}.$

Thus the Maxwell–Ampère law, Gauss's law for magnetism, and Faraday's law, with the aid of (Template:EquationNote), predict the existence of electromagnetic waves together with their speed.

For these reasons, especially the last, one could hardly overstate the importance of identity (Template:EquationNote).

Digression: Proofs from formal products

We have seen that Wilson (1901, pp. 150, 152) interprets the divergence and curl as "formal" or "symbolic" scalar and vector products with the Template:Mathoperator.  Template:Nowrap, in his 1995 report (pp. 26–9), alleges that this interpretation began with Wilson and not with Gibbs. Here I shall submit, on the contrary, that while the terminology may not be attributable to Gibbs, the concept certainly is.

Later in the same report, Tai confuses the picture by citing the first volume of Heaviside's Electromagnetic Theory (1893), where Heaviside, although his notations for the scalar and vector products differ from those of Gibbs, nevertheless considers the Template:Math operator as a factor in such products. Tai continues:

At the time of his writing he [Heaviside] was already aware of Gibbs' pamphlets on vector analysis but Wilson's book was not yet published. It seems, therefore, that Heaviside and Wilson independently introduced the misleading concept for the scalar and vector products between Template:Math and a vector function. Both were, perhaps, induced by Gibbs' notations for the divergence and the curl. Heaviside did not even include the word 'formal' in his description of the products.

Whereas it was quite in character for Heaviside to treat an operator that way, the word "independently" would have surprised Wilson and is contradicted by Tai himself, who observes that Wilson's preface acknowledges Heaviside. In Wilson's own words:

By far the greater part of the material used in the following pages has been taken from the course of lectures on Vector Analysis delivered annually at the University [Yale] by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside's Electromagnetic Theory (Electrician Series, 1893) and in Professor Föppl's lectures on Die Maxwell'sche Theorie der Electricität (Teubner, 1894). ....

Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor.Template:Efn On the other hand there has been a growing tendency especially in the last decade toward the adoption of some form of Vector Analysis. The works of Heaviside and Föppl referred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to completeness has been published. In fact Heaviside says: "I am in hopes that the chapter which I now finish may serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treatment of vectors" (Electromagnetic Theory, Vol. Template:Serif., p. 305). Elsewhere in the same chapter Heaviside has set forth the claims of vector analysis as against Quaternions, and others have expressed similar views.

Most damaging to Tai's thesis, however, is Gibbs's original pamphlet, a copy of which Heaviside received from Gibbs himself in June 1888. Sections 62 to 65 of the pamphlet appear under the heading

Template:Math Template:Math and  Template:Math  applied to Functions of Functions of Position.

In § 62, Gibbs says that a constant scalar factor after such an operator may be placed before it (that is, taken outside the operator). Template:Nowrap he states our rule (Template:EquationNote) for the gradient of a function of a scalar field. His next section (in which I have bolded the vector fieldTemplate:Math) is worth quoting in full:

64.  If Template:Mvar or Template:Math is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of  Template:Math or Template:Math  or Template:Math  will be equal to the sum of the values obtained by making successively all but each one of these variables constant.

This proposition is a generalized product rule in the sense that the "function of several scalar or vector variables" may be, but is not restricted to, any sort of product of those variables. Gibbs continues:

65.  By the use of this principle, we easily derive the following identical equations:

Six "equations" follow. The first says that the gradient operation is distributive over addition, and the second says the same of the divergence and curl (on one line). The last four are our identities (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote), in that order (albeit with different symbols). Gibbs then remarks (with my italics):

The student will observe an analogy between these equations and the formulæ of multiplication. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the Template:Math contains implicitly the vectors Template:Math Template:Math and Template:Math which are to be multiplied  into the following quantities.

Indeed, if the first  factor is constant, identities (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) become

$\begin{array}{cc}\mathrm{\nabla }(p\phi )& =p\mathrm{\nabla }\phi \\ \mathrm{\nabla }\cdot p𝐛& =p\mathrm{\nabla }\cdot 𝐛\\ \mathrm{\nabla }\times p𝐛& =p\mathrm{\nabla }\times 𝐛\\ \mathrm{\nabla }\cdot (𝐚\times 𝐛)& =-𝐚\cdot \mathrm{\nabla }\times 𝐛,\end{array}$

whereas if the second  factor is constant, they become respectively

$\begin{array}{cc}\mathrm{\nabla }(p\phi )& =\phi \mathrm{\nabla }p\\ \mathrm{\nabla }\cdot p𝐛& =\mathrm{\nabla }p\cdot 𝐛\\ \mathrm{\nabla }\times p𝐛& =\mathrm{\nabla }p\times 𝐛\\ \mathrm{\nabla }\cdot (𝐚\times 𝐛)& =\mathrm{\nabla }\times 𝐚\cdot 𝐛.\end{array}$

All eight equations look like rearrangements of products involving a vectorTemplate:Math.  [Concerning the last three equations, we have made that observation before; see (Template:EquationNote) above.]  But only seven of the eight are explained by taking the constant outside the operator (Template:Nowrap); the exception is the fourth, in which the minus sign is not explained by that step alone, but is explained by the change in the cyclic order of the formal triple product. And if we add the two right-hand sides corresponding to each of the four left-hand sides, we get the identities in which both factors are variable—as claimed Template:Nowrap.

If § 65 leaves any doubt that Gibbs approved of formal products with the symbolic vectorTemplate:Math (albeit without using those terms), this is dispelled Template:Nowrap, where he writes:

166.  To the equations in No. 65 may be added many others…

followed by a list of seven identities terminated by "etc." Six of the seven are beyond the scope of the present paper,Template:Efn while the third of the seven is our (Template:EquationNote). After the list comes the smoking gun (Template:Nowrap):

The principle in all these cases is that if we have one of the operators  Template:Math Template:Math Template:Math  prefixed to a product of any kind, and we make any transformation of the expression which would be allowable if the Template:Math were a vector, (viz: by changes in the order of the factors, in the signs of multiplication, in the parentheses written or implied, etc.,) by which changes the Template:Math is brought into connection with one particular factor, the expression thus transformed will represent the part of the value of the original expression which results from the variation of that factor.

The italics are mine, but I have refrained from italicizing those instances of the word "factor" which are not applicable toTemplate:Math. In particular, at the stage when "the Template:Math is brought into connection with one particular factor," the "part of the value… which results from the variation of that factor" evidently means the term of the sum Template:Nowrap —which, as we have noted, amounts to a generalized product rule. But, according to the stated "principle', we reach that stage by treatingTemplate:Math as a factor. I rest my case.

Wilson (1901, p. 157) gives a comprehensive list of sum and product rules for the gradient, divergence, and curl, and properly states (p. 158) that the rules may be proven "most naturally" from Gibbs's definitions of the operators—our equations (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote). Understandably, Wilson uses a Template:Math sign rather than implicit summation. Less understandably, and less fortunately, he does not sum over a numerical index; e.g., he defines the curl operator as

Template:Big

and explains that "The summation extends over Template:Math."  With these definitions he proves our identities (Template:EquationNote) and (Template:EquationNote) essentially as we have done, but inevitably with greater difficulty, which may explain why he then says "The other formulæ are demonstrated in a similar manner" before reverting to Gibbs's strategy of varying one factor at a time. He announces (p. 159) that the variable held constant will be written as a subscript after the product, and he combines this notation with his Template:Math notation in a rigorous proof that varying one factor at a time is valid for our (Template:EquationNote), i.e. the gradient of a dot-product. Noting that this result is analogous to

$d(𝐮\cdot 𝐯)=𝐮\cdot d𝐯+d𝐮\cdot 𝐯,$

he then jumps to the conclusion that varying one factor at a time is valid for all  of his product rules—notwithstanding that a small change in a vector is not related to its divergence or curl as a small change in a scalar is related to its gradient.

That per saltum  conclusion is his cue to go formal and symbolic. To obtain the curl of a cross-product [as in our (Template:EquationNote)], he "formally" expands a vector triple product to obtain the curl when the first factor is constant, states the curl when the second factor is held constant, and adds the two partial curls (Wilson, 1901, p. 161). Next he gives various arrangements of our (Template:EquationNote), except that he presents the first vector not as strictly uniform, but as merely held constant for the gradient operation. He states in passing that a proof may be effected by "expanding in terms of  Template:Math"; but instead of such a proof, he offers a "method of remembering the result" by expanding the "product"  Template:Math  "formally as if  Template:Math  were all real vectors" (pp. 161–2). Concerning the curl of the gradient, and the divergence of the curl (pp. 167, 168), he recommends expanding in terms of  Template:Math  but does not elaborate. Concerning the curl of the curl, however, he shows what would happen if it were "expanded formally according to the law of the triple vector product" (p. 169).

In defense of the "formal product" method, we should note that the operators Template:Math Template:Math and Template:Math are linear, so that they are distributive over addition and may be permuted with multiplication by a constant, as if the operators themselves were multipliers (like components of vectors). They may be similarly permuted with other like operators—explaining why the formal-product method correctly deals with the curl of the gradient, the divergence of the curl, and the curl of the curl. But such an operator cannot be permuted with multiplication by a variable, because then the product rule of differentiation applies, yielding an extra term. The formal-product system responds to this difficulty by generalizing the product rule as in §§ 64 & 166 of Gibbs (1881–84). As Borisenko & Tarapov put it (1968, p. 169),

the operator Template:Math acts on each factor separately with the other held fixed. Thus Template:Math should be written after any factor regarded as a constant in a given term and before any factor regarded as variable.

In this they differ inconsequentially from Gibbs, who requires that the operator be "brought into connection" with the factor considered variable.

To illustrate, let us find the gradient of a dot-product, essentially in the manner of Borisenko & Tarapov (1968, p. 180), quoted by Tai (1995, p. 46; the next five equation numbers are Tai's). In this case the generalized product rule gives Template:NumBlk where the subscript Template:Mvar marks the factor held constant during the differentiation. In Wilson's notation, this equation would be written

Template:Midsize

where a trailing subscript indicates which factor is held constant.  In the Feynman subscript notation, the subscript is attached to the Template:Math operator and indicates which factor is allowed to vary, so that the same equation would be written

Template:Midsize

But, as we are discussing Borisenko & Tarapov, we press on with (Template:EquationNote).  By the algebraic identity Template:NumBlk i.e.

$𝐜(𝐚\mathbf{\cdot }𝐛)=(𝐚\mathbf{\cdot }𝐜)𝐛+𝐚\times (𝐜\times 𝐛),$

we can say Template:NumBlk Similarly, Template:NumBlk Substituting (Template:EquationNote) and (Template:EquationNote) into (Template:EquationNote), in which the order of the dot-products is immaterial, and dropping the Template:Mvarsubscripts (because they are now outside the differentiations), we get the correct result Template:NumBlk corresponding to our (Template:EquationNote).

Tai (1995, p. 47) is unimpressed, asking why we cannot apply (Template:EquationNote) directly to the left side of (Template:EquationNote). The answer to that is obvious: on the left side, the Template:Math operator is applied to a product of two variables, and the variations of both must be taken into account. But there is a harder question which Tai does not ask: in (Template:EquationNote), why can't we have Template:Math instead ofTemplate:Math ? (Or, in terms of Feynman subscripts, why can't we have Template:Math instead ofTemplate:Math?) Because that would make the term vanish? Yes, it would; but, as there is only one variable factor on the left side, why do we need two terms on the right? Because the rule says Template:Math should be written after the constant but before the variable? Yes, but that rule serves the purpose of varying each variable, whereas there is only one variable to vary on the left of (Template:EquationNote). The same issue arises in (Template:EquationNote). We cannot settle the question even by appealing to symmetry. Obviously the right side of (Template:EquationNote), like the left, must be unchanged if we switch Template:Math and Template:Math; and indeed it is. But if the first term on the right of (Template:EquationNote) and of (Template:EquationNote) were to vanish, the necessary symmetry of (Template:EquationNote) would be maintained. And unless I'm missing something, Tai's "symbolic vector" method does not circumvent the problem; Tai's "Lemma 2" (1995, p. 53) is the Gibbs⧸Wilson method of "varying one factor at a time", written with Feynman subscripts attached to the symbolic vector instead of the del operator.Template:Efn

For another example of the same issue, consider the following two-liner offered by Panofsky & Phillips (1962, pp. 470–71) and rightly pilloried by Tai (1995, pp. 47–8):

$\begin{array}{cccc}& \mathrm{\nabla }\times (𝐀\times 𝐁)=(\mathrm{\nabla }\cdot 𝐁)𝐀-(\mathrm{\nabla }\cdot 𝐀)𝐁& & [𝗌𝗂𝖼]\\ & =(\mathrm{\nabla }\cdot {𝐁}_c)𝐀+(\mathrm{\nabla }\cdot 𝐁){𝐀}_c-(\mathrm{\nabla }{\mathbf{\cdot }𝐀}_c)𝐁-(\mathrm{\nabla }\mathbf{\cdot }𝐀){𝐁}_c& & [𝗌𝗂𝖼].\end{array}$

If the first line were right, the authors would hardly bother to continue; but evidently it isn't, because it doesn't begin by "varying one factor at a time". The second line does not follow from the first and includes divergences of constants, which ought to vanish but somehow apparently do not. Let's try again, this time sticking to the rules:

$\begin{array}{cc}& \mathrm{\nabla }\times (𝐀\times 𝐁)=\mathrm{\nabla }\times ({𝐀}_c\times 𝐁)+\mathrm{\nabla }\times (𝐀\times {𝐁}_c)\\ & =(\mathrm{\nabla }\cdot 𝐁){𝐀}_c-({𝐀}_c\cdot \mathrm{\nabla })𝐁+({𝐁}_c\cdot \mathrm{\nabla })𝐀-(\mathrm{\nabla }\cdot 𝐀){𝐁}_c\\ & =𝐀(\mathrm{\nabla }\cdot 𝐁)-𝐁(\mathrm{\nabla }\cdot 𝐀)+(𝐁\mathbf{\cdot }\mathrm{\nabla })𝐀-(𝐀\mathbf{\cdot }\mathrm{\nabla })𝐁,\end{array}$

in agreement with our (Template:EquationNote). Here the first line comes from the generalized product rule, and the third is obtained from the second by rearranging terms and dropping the (now redundant) subscripts. The interesting line is the second, which is obtained from the first by expanding the formal vector triple products. But again, why must we have Template:Math and Template:Math instead of Template:Math and Template:Math which would make the middle two terms vanish? Again symmetry does not give an answer. The right-hand side, like the left, must change sign if we switch Template:Math and Template:Math; but the disappearance of the Template:Math and Template:Math terms would maintain the required (anti)symmetry. Funnily enough, the result would then agree with the incorrect first line given by Panofsky & Phillips (above). But then how would we know that it is incorrect?

The foregoing examples show that "formal product" arguments can be tenuous, even on their own terms. Before these examples, we might have been troubled by the omission of a general proof of the "generalized" product rule. After them, we might wonder whether the rule is even well defined.

I submit, however, that none of this matters. I submit that the popularity of using "formal products" with the del operator, in derivations of vector-analytic identities, is a reaction to the failure of early writers to use indicial notation in the Cartesian definitions of differential operators.Template:Efn The ensuing proliferation of terms in coordinate-based derivations led authors to seek shortcuts through "formal products" when more rigorous but no-less convenient shortcuts could have been taken through indicial notation, especially in combination with implicit summation. Our derivation of the gradient of a dot-product (Template:EquationNote) is shorter than that of Borisenko & Tarapov, and even uses the right-hand sides of their identities (Template:EquationNote) and (Template:EquationNote), but obtains them rigorously with no ambiguity and no Template:Mvarsubscripts. Our derivation of the curl of a cross-product (Template:EquationNote) takes six lines with a single column of "=" signs. Our subsequent formal-product derivation (not to be confused with the attempt of Panofsky & Phillips) seems to take only three lines; but it is only through our earlier indicial derivation that we have any confidence in our result (not to be confused with the result of Panofsky & Phillips). Our other indicial derivations of identities are mostly shorter than the two just mentioned. Having amassed so comprehensive a collection of identities so rigorously with so little effort, I submit that the use of formal products, Wilson subscripts, Template:Mvarsubscripts, and Feynman subscripts for this purpose is a historical aberration, to be deciphered in other people's writings but avoided in one's own.

That being said, it is one thing to conclude, as Tai duly does, that the del-cross and del-dot notations should not be interpreted as products in derivations and proofs, and another thing to allege, as Tai also does (1995, p. 22), that  Template:Math  and Template:Math  are "not compound operators" but only "assemblies", or in other words that "Template:Mathis not a constituent of the divergence operator nor of the curl operator." Against the latter proposition, our equations (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) have been derived, not merely defined, and our derivation of (Template:EquationNote) is as general as we could wish. Moreover, whereas (Template:EquationNote) and (Template:EquationNote) are for Cartesian coordinates, we shall see that they have counterparts in more general coordinates.

General coordinates

From our initial definitions of the differential operators, we derived certain identities, from which we derived expressions for the operators in Cartesian coordinates, from which we derived a comprehensive collection of identities, two of which (the multivariate chain rule, and the curl of the product of a scalar and a vector) will now be useful for expressing the operators in other coordinate systems. Cartesian coordinates are traditionally called Template:Math  which we renamed Template:Mvar  where  Template:Math  respectively. The best-known 3D non -Cartesian coordinate systems are the cylindrical coordinates Template:Math and the spherical coordinates Template:Math; we have already seen Template:Mvar  in the guise of the magnitude of the position vectorTemplate:Math.  But now we want our coordinate system to be as general as possible—with the Cartesian, cylindrical, and spherical systems and many others, and even classes of systems, as special cases.

Natural and dual basis vectors

We shall call our general coordinates Template:Mvar  where  Template:Math;  yes, for reasons which will emerge, we shall write the coordinate index as a Template:Nowrap. But we shall write Template:Mvar  forTemplate:Math  relying on context to distinguish it from the special caseTemplate:Mvar.  By describing the Template:Mvar  as coordinates  we mean two things. First, for some domain of interest, the position vector is a smooth function

$𝐫=𝐫(u^1,u^2,u^3),$

which possesses partial derivatives w.r.t. its arguments. Second, for every position vector in the resulting range, there is only one ordered triplet  Template:Math  so that we can think of each coordinate as

Template:Big

—that is, we can think of each Template:Mvar  as a scalar field, which possesses a gradient.Template:Efn (I say "think of" because Template:Mvar, being obviously dependent on a coordinate system, would not normally be considered a true scalar; but sometimes we need to treat the coordinate system itself as an object under study.)

These two properties of coordinates respectively suggest two simple ways of choosing basis vectors related to the coordinates: we shall define the natural basis vectors as Template:NumBlk and the dual basis vectors as Template:NumBlk (We could normalize the natural basis vectors by dividing them by their magnitudes to obtain unit vectors; but, for the moment, we won't bother.) Just as we may think of each Template:Mvar as a scalar field and inquire after its directional derivative or its gradient or its Laplacian, so we may think of each Template:Math orTemplate:Math as a vector field and inquire after its directional derivative or its curl or its divergence or its Laplacian. (That the curl ofTemplate:Math  is zero  will be especially useful.)

In Cartesian coordinates,  Template:Math and Template:Math are both equal to the unit vectorTemplate:Math; thus, in Cartesian coordinates, the natural basis vectors are their own duals.  In general coordinates,  Template:Math and Template:Math may differ in both direction and magnitude and are not generally unit vectors. Nevertheless, even in general coordinates, there is a simple relation between the natural and dual basis vectors. Consider the dot-product

Template:Big

IfTemplate:Math then Template:Math being in a direction in which Template:Mvar varies while each other Template:Mvar does not, is tangential to a surface of constant Template:Mvar and therefore normal to Template:Math so that the dot-product is zero. But by (Template:EquationNote),

Template:Big

and if we vary Template:Math by varying Template:Mvar while holding each other Template:Mvar constant, we can divide by Template:Mvar and obtain Template:NumBlk Putting the two cases together, we have Template:NumBlk where the right-hand function, known as the Kronecker delta function, is defined by Template:NumBlk Obviously the function is symmetric: the indices Template:MvarandTemplate:Mvar  can be interchanged. If two lists of vectors are related so that the dot-product of the Template:Mvarth vector in one list and the Template:Mvarth in the other isTemplate:Mvar, the two lists are described as reciprocal. Thus the triplets Template:Math and Template:Math are reciprocal bases: the dual basis is the reciprocal of the natural basis and vice versa. Hence, taking the natural basis as a reference, the dual basis is sometimes called "the" reciprocal basis.

In Cartesian coordinates, (Template:EquationNote) becomes

Template:Big

So we have a relation for general coordinates (Template:EquationNote) which is just as simple as its special case for Cartesian coordinates, provided that we use the natural basis for one factor and the dual basis for the other. This will be a recurring pattern.

We have deduced the reciprocity relation (Template:EquationNote) from prior definitions of the natural basis Template:Math and the dual basis Template:Math.  This result has a partial converse, in that a reciprocity relation between bases is enough to define either basis in terms of the other—as we shall see later. But first we proceed to components of vector fields.

Contravariant and covariant components

A coordinate grid is a set of intersecting curves such that on each curve, one coordinate varies while the others are constant. If we could embed such a grid in an elastic medium, and then stretch and rotate the medium, the natural basis vectorsTemplate:Math given by (Template:EquationNote) would stretch and rotate with the medium and with the grid. Accordingly, the natural basis is also called the covariant basis. But according to (Template:EquationNote), the dot-product of a natural basis vector and a dual basis vector is invariant (independent of the coordinate system), so that the variation of one factor compensates  for the variation of the other. So, as the natural basis is "covariant" with the coordinate grid, we say that the dual basis is contravariant. Notice that the Template:Nowrap factor has a Template:Nowrap index (easily remembered because "co  rhymes with low ") whereas the Template:Nowrap factor has a Template:Nowrap index, and that one kind of variation must combine with the other in order to produce an Template:Nowrap result; these will be recurring patterns.

A vector field Template:Math may be expressed in components w.r.t. the natural (covariant) basis as Template:NumBlk with summation, or in components w.r.t. the dual (contravariant) basis as Template:NumBlk with summation. IfTemplate:Math is to be invariant (a true vector, existing independently of the coordinate system), the components must be contravariant in the former case and covariant in the latter, and accordingly are written with superscripts and subscripts respectively. In Cartesian coordinates, the two bases are the same, so that the components w.r.t. the two bases are also the same; that's why, in the above section on Cartesian coordinates, we got away with writing component indices as subscripts. In general coordinates, however, the basis vectors have subscripts and the components have superscripts or vice versa, so that the index of implicit summation appears once as a superscript and once as a subscript.

Taking dot-products of (Template:EquationNote) withTemplate:Math, applying (Template:EquationNote), and noting that only one term on the right is non-zero, we obtain Template:NumBlk Similarly, taking dot-products of (Template:EquationNote) withTemplate:Math yields Template:NumBlk These results depend on the reciprocity relation (Template:EquationNote) but not on the earlier definitions of the bases to which that relation applies. They say:

• to find the contravariant components of a vector, take its dot-products with the contravariant basis vectors, and
• to find the covariant components of a vector, take its dot-products with the covariant basis vectors;

or, in terms of the bases themselves:

• to find the components of a vector w.r.t. either basis, take dot-products of that vector with the other basis.

If a particular Template:Mvar has a particular name, such asTemplate:Mvar orTemplate:Mvar, then, if we're not using indexed summation, we may find it convenient to write that name in place of the indexTemplate:Mvar  in the superscript or subscript.

At the present level of generality, the basis vectors Template:Math unlike their Cartesian counterparts Template:Math are not  assumed to be uniform (i.e., homogeneous). One consequence of this general non-uniformity (inhomogeneity) is that, although we can say  Template:Math  in Cartesian coordinates and  Template:Math  in general coordinates, we cannot say

Template:Big

in general coordinates. For example, we have seen that in spherical coordinates the position vector Template:Math is simply Template:Math;  it is not  Template:Math because Template:Mvar and Template:Mvar are encoded in the direction ofTemplate:Math.  Similarly, in cylindrical coordinates the position vector Template:Math is Template:Math;  it is not  Template:Math because Template:Mvar is encoded in the direction ofTemplate:Math.  In both examples, encoding one coordinate in the direction of another coordinate's unit vector is circular in that the said direction depends on the position vector, which is the very thing that we want to represent.

A non-uniform basis is not a global  basis. It cannot give a uniform representation of a uniform vector field, because the standard of representation changes; it is like having a compass whose orientation varies from place to place and⧸or a measuring stick whose length varies from place to place. But it can serve as a local basis —as in (Template:EquationNote) and (Template:EquationNote), each of which expresses a vector field at a given location in terms of a basis at that location, notwithstanding that the basis may be different at other locations. And although a local basis (as we have just seen) cannot generally represent the position vector in a non-circular manner, it can  represent a change  in the position vector. By the generality of the multivariate chain rule (Template:EquationNote),

Template:Big

Multiplying by Template:Mvar we get Template:NumBlk or, substituting from (Template:EquationNote), Template:NumBlk Thus the small changes in the coordinatesTemplate:Mvar  are the components of the true vectorTemplate:Math w.r.t. the covariant  basis. That means the changes in the coordinates must be contravariant. Here at last is the explanation why we write general coordinates with superscript indices. And again the point is moot for Cartesian coordinates, for which the covariant basis is also contravariant.

Since Template:Mvar  is contravariant,  Template:Math  in (Template:EquationNote) must be covariant in order to yield the true vectorTemplate:Math. This vindicates our decision to write Template:Mvar with a subscript. Recall, however, that Template:Mvar  meansTemplate:Math. Thus the derivative w.r.t. the contravariant quantity is covariant —wherefore it is said that a superscript in the denominator of a derivative counts as a subscript in the derivative as a whole.

In (Template:EquationNote), the general term  Template:Math (not the sum) is the displacement ofTemplate:Math due to the small change Template:Mvar in the coordinate Template:Mvar. The three such displacements ofTemplate:Math make concurrent edges of a parallelepiped whose signed volume is

$dV={𝐡}_{1}d{u}^1\cdot {𝐡}_{2}d{u}^2\times {𝐡}_{3}d{u}^3;$

that is, Template:NumBlk where

$J:={𝐡}_1\cdot {𝐡}_2\times {𝐡}_3$

or, to use a standard abbreviation for the scalar triple product, Template:NumBlk Template:Mvar  is called the Jacobian of the natural (covariant) basis. We describe the basis and the associated coordinate system as right-handed if this Jacobian is positive, and left-handed if this Jacobian is negative. Thus the handedness depends on the standard order in which we write the vectors; e.g., the standard Cartesian basis is right-handed because we write it asTemplate:Math but would be left-handed if we wrote it asTemplate:Math.

If the covariant basis is indeed a basis, its member vectors must not be coplanar; that is, Template:Mvarmust not be zero. Hence, if the covariant basis is to be a local basis in some region of interest, Template:Mvarmust not vanish anywhere in that region, and therefore must have the same sign throughout the region; that is, the handedness of the coordinate system must be the same throughout the region.

Properties of reciprocal bases

We have noted that formulae (Template:EquationNote) and (Template:EquationNote), for the components of a vector w.r.t. the covariant and contravariant bases, depend only on the reciprocity relation (Template:EquationNote) between the bases. Now, retaining the designations "covariant" and "contravariant" for convenience, let us see what else we can deduce from that relation.

Most obviously, the reciprocity relation leads to a simple component-based expression for the dot-product of two vector fields, say Template:Math andTemplate:Math provided that we use the contravariant components and covariant basis (Template:EquationNote) for one vector, and the covariant components and contravariant basis (Template:EquationNote) for the other:

Template:Big

whence selecting the non-zero terms gives Template:NumBlk And the two vectors, being general, can swap roles in (Template:EquationNote) and (Template:EquationNote): Template:NumBlk

The cross-product needs a bit more preparation. First we define the permutation symbol Template:Mvar orTemplate:Mvar (also called the Levi-Civita symbol) as having the value  Template:Math if Template:Math is a permutation ofTemplate:Math in the same cyclic order,  Template:Mathif Template:Math is a permutation ofTemplate:Math in the reverse cyclic order, and Template:Math if Template:Math is not a permutation, i.e. if there is at least one repeated index. To put it more formally, Template:NumBlk Note that because switching any two indices changes the cyclic order, switching any two indices changes the sign of the permutation symbol. Now by (Template:EquationNote),  Template:Mathis perpendicular to both Template:MathandTemplate:Math. So we can say

${𝐡}_2\times {𝐡}_3={\alpha }_1{𝐡}^1$

where Template:Mathis a real variable to be determined. Taking dot-products withTemplate:Math and applying (Template:EquationNote) and (Template:EquationNote), we find that  Template:Math so that Template:NumBlk By the generality of the vectors we can rotate the three indices, but the sign of the left-hand side changes if we swap the two indices on the left. All six cases are covered by Template:NumBlk Here we want only one term; but we need not specify "no sum", because for given Template:MvarandTemplate:Mvar  the permutation symbol leaves only one non-zero term in the sum overTemplate:Mvar. In words, this result says that the cross-product of two covariant basis vectors, with their indices in the standard cyclic order, is the Jacobian times the contravariant basis vector with the omitted index. Similarly, or rather reciprocally, Template:NumBlk where Template:Mvar  is the Jacobian of the contravariant basis.

Equations (Template:EquationNote) and (Template:EquationNote), which we have obtained from the reciprocity relation (Template:EquationNote), can be solved for Template:MathandTemplate:Math respectively; but now we do suppress the implicit sum, because Template:Mvar  is "given" instead of Template:MvarandTemplate:Mvar: Template:NumBlk Template:NumBlk Thus a reciprocity relation between bases is enough to define either basis in terms of the other—as claimed above. If it is not convenient to suppress an implicit sum, the last two results can instead be written Template:NumBlk and Template:NumBlk where the factor 2 in each denominator is needed because the right-hand side has two equal non-zero terms—the sign of the permutation symbol compensating for the order of the cross-product.

Now we're ready to consider the cross-product of two vector fields. In terms of the covariant basis,

Template:Big

i.e., Template:NumBlk On the right, the two components and the basis vector are contravariant, but invariance is achieved by multiplying by the covariant Jacobian (which has three covariant factors). Similarly, Template:NumBlk On the right of (Template:EquationNote) or (Template:EquationNote), the implicit triple summation has 27 terms, of which only six—corresponding to the six possible permutations of the three possible indices—can be non-zero. Thus the factor following the Jacobian can be recognized as the familiar determinant whose columns (or rows), in cyclic order, are the components ofTemplate:Math the components ofTemplate:Math and the three basis vectors. In Cartesian coordinates, in which the Jacobians are equal toTemplate:Math and we don't need the co⧸contra distinction, both equations reduce to

Template:Big

—a familiar result written in a possibly unfamiliar way.

The Jacobian of the contravariant basis is

$J^{\prime }={𝐡}^1\cdot {𝐡}^2\times {𝐡}^3$

or, if we substitute from (Template:EquationNote),

$\begin{array}{cc}{J}^{\prime }& =\frac{({𝐡}_2\times {𝐡}_3)}{J}\cdot \frac{({𝐡}_3\times {𝐡}_1)}{J}\times \frac{({𝐡}_1\times {𝐡}_2)}{J}\\ & =\frac{({𝐡}_2\times {𝐡}_3)\cdot ({𝐡}_3\times {𝐡}_1)\times ({𝐡}_1\times {𝐡}_2)}{J^3}.\end{array}$

In the numerator, the cross-product of cross-products can be read as a vector triple product in which the first factor is a cross-product. Expanding that triple product and noting that one term is a scalar triple product with a repeated factor, we get

$J^{\prime }=\frac{({𝐡}_2\times {𝐡}_3)\cdot J{𝐡}_1}{J^3}=\frac{J^2}{J^3}=\frac{1}{J},$

so that we may write Template:NumBlk in (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote). In words, the Jacobian of the reciprocal basis is the reciprocal of the Jacobian of the original basis. Therefore the two Jacobians have the same sign. Therefore a basis is right-handed if and only if its reciprocal is right-handed. Thus the natural and dual bases of a coordinate system have the same handedness, and the handedness of either may be identified with the handedness of the coordinate system.

The gradient, del, and advection operators

Let Template:Mvar  be a scalar field, and letTemplate:Mvar  be arc length in the direction of the unit vectorTemplate:Math. By the multivariate chain rule (Template:EquationNote),

Template:Big

So  Template:Math  is the vector whose (invariant) scalar component in the direction of anyTemplate:Math is the directional derivative ofTemplate:Mvar in that direction; that is, Template:NumBlk or, in operational terms, Template:NumBlk Apart from the need to pair a superscript with a subscript, these two results look as simple as their Cartesian special cases (Template:EquationNote) and (Template:EquationNote).

If Template:Mvar  is a generic field and Template:Math is a general vector in the direction of the sameTemplate:Math then by definition (Template:EquationNote),

Template:Big

that is, Template:NumBlk or, in operational terms, Template:NumBlk These results likewise look as simple as their Cartesian special cases (Template:EquationNote) and (Template:EquationNote).  And by (Template:EquationNote), the Template:Math operator again turns out to be the formal dot-product of  Template:Math andTemplate:Math.

The curl and divergence operators

To express the curl of a vector fieldTemplate:Math we choose the contravariant basis (Template:EquationNote) and apply identity (Template:EquationNote):

Template:Big

On the right, the first term vanishes because Template:Math  is Template:Math (and the curl of a gradient is zero). Substituting from (Template:EquationNote) in the second term, we obtain

Template:Big

or, using (Template:EquationNote), Template:NumBlk or, in a more familiar form,

$\mathrm{curl}𝐪=J^{\prime }\left|\begin{array}{ccc}{𝐡}_1& {\partial }_1& q_1\\ {𝐡}_2& {\partial }_2& q_2\\ {𝐡}_3& {\partial }_3& q_3\end{array}\right|.$

Formula (Template:EquationNote) agrees with a result obtained by Tai with his "symbolic vector" method. It is also what we would get by naively using (Template:EquationNote) to evaluate Template:Math  it comes out so simply because each contravariant basis vectorTemplate:Math  is the actual gradient ofTemplate:Mvar and not (e.g.) merely a unit vector in the same direction (remember that Template:Mvar is the component w.r.t.Template:Math  notTemplate:Math).

But (Template:EquationNote) does not end in a subexpression for the operandTemplate:Math  and therefore does not directly yield an expression for the curl operator. To find this operator and the divergence operator, we return to the original definitions (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote), noting that they can be combined as Template:NumBlk where Template:Math may be a null for the gradient, a cross for the curl, or a dot for the divergence.Template:Efn Recalling that the value of this expression does not depend on the shape ofTemplate:Mvar, letTemplate:Mvar be the parallelepiped defined by the six equicoordinate surfaces at Template:MvarandTemplate:Mvar, so that Template:Mvar  is given by (Template:EquationNote). Then the contribution to the integral from the face atTemplate:Mathis

$\left[\right({𝐡}_{2}d{u}^2\times {𝐡}_{3}d{u}^3)*\psi {]}_{u^1+d{u}^1}$

where the square brackets and subscripting mean "evaluated at". This can be written

$d{u}^{2}d{u}^3[({𝐡}_2\times {𝐡}_3)*\psi {]}_{u^1+d{u}^1}$

or, by (Template:EquationNote),

$d{u}^{2}d{u}^3[J{𝐡}^1*\psi {]}_{u^1+d{u}^1}.$

Similarly, the contribution from the face atTemplate:Math (where Template:Math points inward instead of outward) is

$-d{u}^{2}d{u}^3[J{𝐡}^1*\psi {]}_{u^1}.$

The sum of the contributions from the two opposite faces can then be written

$d{u}^{1}d{u}^{2}d{u}^3{\partial }_1(J{𝐡}^1*\psi ),$

so that when we add in the contributions from the other two pairs of opposite faces, the entire integral becomes

Template:Big

(with implicit summation overTemplate:Mvar). Substituting this and (Template:EquationNote) into (Template:EquationNote), we get Template:NumBlk Now applying the product rule gives Template:NumBlk Here the left-hand side is Template:Math  according to our original volume-based definition (Template:EquationNote) of the Template:Mathoperator—which is known to yield the curl or the divergence if  Template:Math is a cross or a dot, respectively—whereas the first term on the right is what we would get for Template:Math  by using our latest definition (Template:EquationNote) of the Template:Mathoperator and allowingTemplate:Mvar  to "pass by" the star in the Wilsonian manner. So, if we can show that the second term on the right is zero, we shall have established the precise sense in which the del-cross and del-dot notations are valid in general coordinates. In that second term, by (Template:EquationNote),

Template:Big

where the last line follows by (Template:EquationNote). But the order of partial differentiation can be switched. So, in the sum over the permutations, for each term inTemplate:Math  there is an equal term inTemplate:Math  to which the permutation symbol attaches the opposite sign, so that the terms inTemplate:Math  cancel. Similarly the terms inTemplate:Math  cancel. Thus, as anticipated, the second term in (Template:EquationNote) is zero and we have Template:NumBlk

If  Template:Math is a null and Template:Mvar is a scalar field Template:Math then (Template:EquationNote) becomes (Template:EquationNote) and thus (fortunately!) confirms (Template:EquationNote) as the form of the del operator in general coordinates.

Now let Template:Mvar be a vector field Template:Math.  IfTemplate:Math is a cross, then (Template:EquationNote) becomes Template:NumBlk or, in operational terms, Template:NumBlk If instead Template:Math is a dot, (Template:EquationNote) becomes Template:NumBlk or, in operational terms, Template:NumBlk But if we take the Template:Math operator as given by (Template:EquationNote) and try to construct the curl and divergence operators (in the same  coordinates) as Template:Math  and Template:Math  respectively, we get  Template:Math  and  Template:Math  respectively [compare (Template:EquationNote) and (Template:EquationNote)]; and if we then letTemplate:Mvar  "pass by" the cross and the dot, we get (Template:EquationNote) and (Template:EquationNote), or (Template:EquationNote) and (Template:EquationNote) if we include the operandTemplate:Math. Thus the del-cross and del-dot notations work in general coordinates.

Equations (Template:EquationNote) to (Template:EquationNote) are apparently due to Tai (1995, eqs. 9.39, 9.40, 9.34, & 9.35, and text on p. 66), who derives them, along with the corresponding form of the del operator (his eq. 9.33), from volume-based definitions expressed in his "symbolic vector" notation. But he does not point out that the curl and divergence operators are obtainable from that del operator, as del-cross and del-dot, via the same "pass by" step that he condemns in the Cartesian context. Speaking of which, we should note that our equations (Template:EquationNote) to (Template:EquationNote), apart from the need to pair a superscript with a subscript, are as simple as their Cartesian special cases (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote).

In (Template:EquationNote) and (Template:EquationNote), it goes without saying that  Template:Math  must be evaluated correctly—in particular, that if the operand is expressed in terms of non-uniform basis vectors, the non-uniformity must be taken into account. Formula (Template:EquationNote), for the curl, does not suffer from this complication, because it is already expressed in components w.r.t. the contravariant basis (whose non-uniformity has already been taken into account). To obtain a similarly convenient formula for the divergence, we use components w.r.t. the Template:Nowrap basis (i.e., contravariant components): in (Template:EquationNote), ifTemplate:Math is a dot and Template:Mvar is a vector fieldTemplate:Math we have

Template:Big

or, by (Template:EquationNote), Template:NumBlk This too agrees with Tai (1995, p. 65, eq. 9.37).

The Laplacian

For a scalar operand, applying (Template:EquationNote) and reversing the "pass by", we find that the Laplacian operator is

Template:Big

And by the linearity of the Laplacian, theTemplate:Math formulation remains valid if the operand is a fixed linear combination of scalars—including a vector field, because that is expressible (even if not actually expressed) w.r.t. a uniform basis. (And if it is expressed in terms of a non-uniform basis, the non-uniformity must be taken into account in differentiations.)

In what follows, however, we shall find it convenient to take a different approach. IfTemplate:Mvar  is a scalar field, its gradient as given by (Template:EquationNote) is  Template:Math of which the Template:Mvarth contravariant component isTemplate:Math which takes the place ofTemplate:Mvar in (Template:EquationNote), so that the divergence of the gradient ofTemplate:Mvaris Template:NumBlk This remains well-defined ifTemplate:Mvar  is a generic field (although we still need to deal with any non-uniformity of the basis in whichTemplate:Mvar  might be expressed).

Affine coordinates

If a basis is uniform (homogeneous), so is its Jacobian. Hence, by (Template:EquationNote) and (Template:EquationNote), the dual (contravariant) basis is uniform if and only if the natural (covariant) basis is uniform. A coordinate system in which these bases are uniform is described as affine. In affine coordinates,

• by (Template:EquationNote), Template:Mathis uniform, so that the curves on which only one coordinate varies are straight parallel lines; and
• by (Template:EquationNote), Template:Mathis uniform, so that the level surfaces of each coordinate (being perpendicular to Template:Math) are parallel planes.

Obviously Cartesian coordinates are affine; but one can also construct affine coordinate systems in which the three vectors of each basis are not mutually perpendicular and⧸or the coordinates have different scales or different units.

We have noted above that the correct application of the del-cross, del-dot, and del-squared notations must allow for non-uniformity of the basis vectors. Obviously this issue does not arise in affine coordinates, including Cartesian coordinates. Hence, while these notations are not (as is sometimes alleged) invalid in other coordinate systems, it would be fair to say that they are safer and more convenient in affine coordinates, including Cartesian coordinates.

Orthogonal coordinates

We know, e.g. from (Template:EquationNote) and (Template:EquationNote), that if two bases are reciprocal, the cross-product of the Template:Mvarth and Template:Mvarth members of one basis is collinear with the Template:Mvarth member of the other, if Template:Math  are distinct. But if the first basis is orthogonal (that is, if its three member vectors are mutually orthogonal), the same cross-product is also collinear with the Template:Mvarth member of the same basis, so that corresponding members of the two bases are collinear. It follows that the natural basis of a coordinate system is orthogonal if and only if the dual basis is orthogonal. And if the bases are orthogonal, the coordinate system itself is said to be orthogonal.

Cartesian coordinates are obviously both affine and orthogonal, and we have already implied that there is a class of coordinate systems that are affine but not orthogonal. The most widely-used class of non-Cartesian systems, however, contains the systems that are orthogonal but not affine; this class, of which the cylindrical and spherical systems are the best-known members, is the class of curvilinear orthogonal coordinates. But we shall drop the word curvilinear  in order to include Cartesian coordinates as a special case.

In orthogonal coordinates, expressing a member of one basis in terms of its reciprocal basis is especially simple because corresponding members of the two bases are collinear, wherefore we can say

Template:Big

where Template:Mvar is a real variable to be determined (and the single index on the left-hand side means no summation). Substituting this into (Template:EquationNote) gives

Template:Big

where Template:NumBlk so that Template:NumBlk And substituting that into (Template:EquationNote), and comparing the result with (Template:EquationNote), we get Template:NumBlk Comparing (Template:EquationNote) with definition (Template:EquationNote), we see that Template:Mvar is the magnitude ofTemplate:Math. Accordingly Template:Mvar is called the scale factor associated with the coordinateTemplate:Mvar; it is the factor by which we multiply a small change inTemplate:Mvar to obtain the magnitude of the consequent change in position.Template:Efn

If we now define Template:NumBlk then, due to the orthogonality,  (Template:EquationNote) and (Template:EquationNote) are respectively reduced to Template:NumBlk and Template:NumBlk —although, for brevity, we shall sometimes leave things in terms ofTemplate:Mvar.

At this point, we could  substitute (Template:EquationNote) and (Template:EquationNote) into earlier equations and obtain a suite of formulae for the differential operators in terms of the covariant basis and Template:Nowrap components! But we can avoid this confusing breach of convention by normalizing the basis vectors.

An orthonormal basis is one whose members are mutually orthogonal unit vectors. The assumption of unit vectors is introduced so late because it is more useful with orthogonality than without. If one basis consisted of unit vectors that were not all orthogonal, then the reciprocal basis vectors given by (Template:EquationNote) or (Template:EquationNote) would not all be unit vectors.Template:Efn But if the basisTemplate:Math consists of orthogonal unit vectors, equation (Template:EquationNote) implies that the reciprocal basis consists of the same vectors; and the converse is also true, by the symmetry of the reciprocity relations. Thus an orthonormal basis is its own reciprocal. Hence, if we choose an orthonormal basis, we do not need superscripts to distinguish the reciprocal basis from the original, or to distinguish components w.r.t. the latter basis from those w.r.t. the former.

An orthonormal basis is not generally covariant, because it doesn't stretch with the coordinate grid (although it does rotate with the grid). Neither is it generally contravariant, because its reciprocal (i.e. itelf) is not generally covariant. Hence, if a non -orthonormal natural or dual basis of an orthogonal coordinate system is normalized (replaced by unit vectors in the same directions), the resulting orthonormal basis is not covariant or contravariant, and components with respect thereto are not contravariant or covariant, and the new basis vectors are not given in terms of the coordinates by (Template:EquationNote) or (Template:EquationNote); the basis is therefore described as a non-coordinate basis. By default, the indices of the orthonormal basis vectors and associated components are written as subscripts, but these are not indicative of covariance. The coordinates themselves remain contravariant (e.g., if the grid dilates, the same movement in space corresponds to smaller changes in the coordinates); but, for want of covariant basis vectors to pair them with, we tend to write the coordinates with subscripts when the basis is orthonormal.

Nevertheless, it is convenient to have one basis instead of two. Moreover, the components of a vector w.r.t. an orthonormal basis are physical components: they have the same dimension (same units) as the represented vector, and they are the components that we would have in mind if we wanted to measure the "components" in the directions of the basis vectors. Hence an orthonormal basis is called a physical basis. Accordingly, it is indeed common practice to normalize the basis vectors of orthogonal coordinate systems. This together with the prevalence of such coordinate systems helps to account for the familiarity of subscripts as indices, and for the jarring unfamiliarity of superscript indices when general (possibly non-orthogonal) coordinates are encountered for the first time.

To normalize the covariant basis, letTemplate:Math (as usual) be the unit vector in the direction ofTemplate:Math Then, by (Template:EquationNote), Template:NumBlk (again with no summation, due to the single index on the left). Hence (Template:EquationNote) becomes: Template:NumBlk A vector field Template:Math is expressed in components w.r.t. the basisTemplate:Math as Template:NumBlk (with summation), where Template:NumBlk (Here the hat onTemplate:Math is needed to distinguish the coefficient ofTemplate:Math from the coefficient ofTemplate:Math and indicates thatTemplate:Math is the coefficient of a unit vector—not thatTemplate:Math has unit magnitude.) TakingTemplate:Mvar as given by (Template:EquationNote) and applying (Template:EquationNote) and (Template:EquationNote), we get Template:NumBlk whence (Template:EquationNote) gives Template:NumBlk

Equation (Template:EquationNote) quantifies the non-covariance of the orthonormal basis; substituting (Template:EquationNote) into (Template:EquationNote), we find that the components ofTemplate:Math with respect toTemplate:Math  are not simplyTemplate:Math butTemplate:Math  [no sum]. So, as the coordinatesTemplate:Mvar are still contravariant, the orthonormal basis vectorsTemplate:Math are not covariant unless the scale factorsTemplate:Mvar  are equal toTemplate:Math — that is, unless the coordinates are Cartesian (except possibly for the handedness). And in Cartesian coordinates we can use subscripts throughout. This is another reason why, when using an orthonormal basis, we might as well write the coordinates asTemplate:Math.

We can now re-express dot- and cross-products w.r.t. the orthonormal basisTemplate:Math. If we apply (Template:EquationNote) and (Template:EquationNote) in (Template:EquationNote) or (Template:EquationNote), the scale factors cancel and we are left with Template:NumBlk as if the coordinates were Cartesian. And if we apply (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) in (Template:EquationNote), the product of the scale factors cancels and we are left with

Template:Big

again as if the coordinates were Cartesian, except that the handedness symbol Template:Mvar  gives a change of sign for left-handed coordinates. The last result is confirmed by applying (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) in (Template:EquationNote). It can also be written Template:NumBlk

We can similarly re-express the first-order differential operators. Applying (Template:EquationNote) in (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) gives respectively Template:NumBlk Template:NumBlk and Template:NumBlk And applying (Template:EquationNote) in (Template:EquationNote) and (Template:EquationNote) gives Template:NumBlk and Template:NumBlk And applying (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) in (Template:EquationNote) gives

Template:Big

or, in determinant form, Template:NumBlk

For the Laplacian, applying (Template:EquationNote) twice in (Template:EquationNote) gives

Template:Big

where the parenthesized dot-product is simplyTemplate:Mvar.  Selecting the non-zero terms, we are left with Template:NumBlk

Working entirely within the coordinatesTemplate:Math we can use equations (Template:EquationNote), (Template:EquationNote) to (Template:EquationNote), and (Template:EquationNote) for scalarTemplate:Math provided that we know the scale factors in terms ofTemplate:Mvar.Template:Efn  And we can find the scale factors in terms ofTemplate:Mvar  if we know the Cartesian coordinatesTemplate:Mvar  in terms ofTemplate:Mvar.  For then the position vector can be written

Template:Big

whence

Template:Big

so that the scale factors can be found from

Template:Big

In (Template:EquationNote) to (Template:EquationNote), the hat onTemplate:Math was needed because we treated the orthonormal basis as a special case, having used a hatlessTemplate:Mvar  in less special cases; the hat would not have been needed if we had assumed an orthonormal basis at the outset. In more elementary introductions to curvilinear orthogonal coordinates, the basis vectors are indeed chosen as unit vectors and consequently as orthonormal vectors. Hence, if the coordinates are called$u,v,w$ and the respective basis vectors are called${𝐞}_u,{𝐞}_v,{𝐞}_w$ (understood to be unit vectors), the components of the vectorTemplate:Math w.r.t. that basis are called$q_u,q_v,q_w,$ with no hats. In this notation, in which sums are written out longhand without numerical indices, it is convenient also to write out the Jacobian in full, in order to exploit cancellations of scale factors. If the Jacobian appears in both a numerator inside parentheses and a denominator outside, the handedness symbolTemplate:Mvar  also cancels. Thus the equations numbered (Template:EquationNote) to (Template:EquationNote) can be rewritten as, respectively, Template:NumBlk Only in the cross-product and the curl does the handedness factorTemplate:Mvar  make any difference. If the system is right-handed—as is also often assumed at the outset—this factor is replaced byTemplate:Math.

For some readers, equation group (Template:EquationNote) will announce a return to familiar territory. For the writer, it offers a convenient place to stop.

Appendix: Mathematizing Huygens' principle

If a wavelike disturbance originating outside  a regionTemplate:Math bounded by a surfaceTemplate:Math enters the region, it must do so through the surfaceTemplate:Mvar.  Unless we believe in "action at a distance", we must conclude that the behavior of the wave function throughout the region is fully determined by its behavior on the bounding surface. That reasoning, being qualitative, does not tell us precisely what aspects of the behavior at the boundary determine the behavior throughout the region, or how. In this appendix, we shall answer these questions using tools of vector analysis—some of which we shall find ready-made, and some of which require assembly. Our aim is to express the wave function in the regionTemplate:Mvar as a surface integral, over the bounding surfaceTemplate:Math of an integrand related to the wave function incident at a general point on that surface.

Huygens' principle asserts not only that the behavior of the wave function throughout the region (containing no sources) is determined by the behavior at the boundary, but also that the behavior at the boundary is equivalent to a distribution of sources over the boundary, so that the wave function throughout the region is as if  the original sources outside the region (primary sources) were replaced  by sources distributed over the boundary (secondary sources).Template:Efn We shall find that by appropriately arranging the integrand for the wave function inside the region, we can indeed recognize the distribution of boundary sources that would generate the wave function. We shall also find that the distribution of secondary sources has an alternative description which is especially convenient in the case of a single monopole primary source.

Hints

If the surface integrand represents a secondary source density, it will not only be related to the primary wave function at a general point onTemplate:Mvar, but will also be delayed by the propagation time from that point to the observation point, and attenuated in accordance with the propagation distance. Hence, ifTemplate:Math is the primary wave function and Template:Math is the position of the observation point (field point) at a distanceTemplate:Mvar  from positionTemplate:Math then the integrand (or at least the dominant term thereof) should be proportional to Template:NumBlk where the square brackets indicate that the contents are to be delayed (or, in older literature, "retarded") byTemplate:Math  relative to the default argumentsTemplate:Math, so that Template:NumBlk

Of course we would like our distribution of secondary sources to be valid for an arbitrarily shaped boundaryTemplate:Mvar. This preference will be easier to satisfy if the distribution of secondary sources, by itself, produces a zero wave function outsideTemplate:Mvar —in other words, no backward secondary waves —because in that case, even if Template:Mvar  is concave outward, the wave function insideTemplate:Mvar  will not be complicated by "backward" waves generated at one point onTemplate:Mvar  and enteringTemplate:Mvar  through another point onTemplate:Mvar.  Accordingly, we would like our surface integral to be equal to the volume  integral overTemplate:Mvar  of  Template:Math because that volume integral will be Template:Math  if Template:Math is insideTemplate:Mvar, but zero if it is outside. In the integrand, the first factor can be replaced byTemplate:Math because the delta function selects the positionTemplate:Math where the propagation distanceTemplate:Math  is zero. Moreover, by (Template:EquationNote), that delta function is the Laplacian ofTemplate:Math.  So the desired integrand, multiplied byTemplate:Math, becomes Template:NumBlk Relating the volume integral of (Template:EquationNote) to the surface integral of something like (Template:EquationNote) would seem to require a surface-to-volume integral identity involving two different fields. Three such identities, all named for George Green, come to mind.

Green's identities

If$u$ and$v$ are scalar fields, then by identity (Template:EquationNote),

$\mathrm{d}\mathrm{i}\mathrm{v}(u\mathrm{\nabla }v)\equiv u\mathrm{△}v+\mathrm{\nabla }u\cdot \mathrm{\nabla }v.$

Integrating both sides over a volumeTemplate:Math enclosed by a surfaceTemplate:Math and applying the divergence theorem on the left, we get

$\underset{S}{\iint }u\mathrm{\nabla }v\cdot \widehat{𝐧}dS\equiv \underset{V}{\iiint }(u\mathrm{△}v+\mathrm{\nabla }u\cdot \mathrm{\nabla }v)dV,$

where Template:Math is the unit normal toTemplate:Mvar  pointing out ofTemplate:Mvar. This integral equation is called Green's first identity. Switching the roles of$u$ and$v$ yields a second integral equation, which can be subtracted from the first to obtain

$\underset{S}{\iint }(u\mathrm{\nabla }v-v\mathrm{\nabla }u)\cdot \widehat{𝐧}dS\equiv \underset{V}{\iiint }(u\mathrm{△}v-v\mathrm{△}u)dV;$

this is Green's second identity. If Template:Mvar is the normal distance fromTemplate:Mvar (positive outsideTemplate:Math negative inside), then, by relation (Template:EquationNote) between the gradient and the directional derivative, we can rewrite Green's second identity in the alternative form Template:NumBlk which remains meaningful if one of the two operands is a generic field. And indeed, by the linearity of the various operators, the identity remains valid in that case (which is not always pointed out).

If, taking the above "hints", we put  $u=1/s$ and  $v=[\psi ]$ in (Template:EquationNote), we get

$\begin{array}{cc}\underset{S}{\iint }& (\frac{1}{s}{\partial }_n[\psi ]-[\psi ]{\partial }_n\frac{1}{s})dS\\ & =\underset{V}{\iiint }\frac{1}{s}\mathrm{△}[\psi ]dV-\underset{V}{\iiint }[\psi ]\mathrm{△}\frac{1}{s}dV,\end{array}$

which is a particular case of what is sometimes called Green's third identity (although that label has also been attached to other things). Applying the chain rule to the second term in the surface integrand, and recalling the purpose of (Template:EquationNote), we can rewrite this result as Template:NumBlk

Kirchhoff's integral theorem

In (Template:EquationNote), the first term in the surface integrand is dominant for sufficiently largeTemplate:Mvar  and, as hoped, this dominant term is proportional to (Template:EquationNote), because Template:Math is proportional toTemplate:Math. The second volume integral, as hoped, isTemplate:Math  if Template:Math is insideTemplate:Mvar, but zero if it is outside. If we can somehow convert the first volume integral to a surface integral, a trivial rearrangement will express the second volume integral as a surface integral, as hoped. And from the terms that we know so far, the surface integral looks promising.

In that first term, however, the factor Template:Math is not to be confused withTemplate:Math; the latter holdsTemplate:Mvar  constant while Template:Mvarvaries, whereas the former also accounts for the variation ofTemplate:Mvar  withTemplate:Math (and consequentlyTemplate:Math) at a fixedTemplate:Math. From (Template:EquationNote), the contribution toTemplate:Math through the variation ofTemplate:Mvar  is

$\frac{\partial \psi (t-s/c)}{\partial (t-s/c)}\frac{\partial (t-s/c)}{\partial s}\frac{\partial s}{\partial n},$

i.e.

Template:Big

Putting the pieces together, we get Template:NumBlk Similarly, if Template:Mvar  denotesTemplate:Mvar  in Cartesian coordinates, Template:NumBlk The last result can be applied three times to the volume integrand in (Template:EquationNote); its applications are shown in Template:Color in the following sequence, which uses Cartesian coordinates with implicit summation:

Template:Big

i.e., Template:NumBlk

To proceed further, we need to assign a direction toTemplate:Math which we have defined as the propagation distance from Template:MathtoTemplate:Math. Being a distance, Template:Mvaris positive whether we measure it forwards or backwards w.r.t. the direction of propagation. But because Template:Math for present purposes, is fixed while Template:Math varies, we shall measureTemplate:Math "backwards", i.e. from Template:MathtoTemplate:Math with  Template:Math  as the displacement vector from Template:MathtoTemplate:Math (so that we can think of both Template:MvarandTemplate:Mvar  as coordinates of positionTemplate:Math, and think ofTemplate:Mvar  as a scalar field). Under that convention, Template:NumBlk so that  Template:Math.  And because Template:Math like Template:Math  is the distance from an origin,  $\mathrm{△}s$ can be evaluated from (Template:EquationNote):

Template:Big

With these substitutions, (Template:EquationNote) becomes Template:NumBlk

If$\psi$  is a scalar  field, we can put this in a more convenient form. By identity (Template:EquationNote),

Template:Big

i.e., by (Template:EquationNote),

Template:Big

i.e., if we apply identity (Template:EquationNote) in the first term on the right, and use components in the second,

Template:Big

(with summation overTemplate:Mvar), whereTemplate:Mvar  is the Template:Mvarth component ofTemplate:Math.  But  Template:Math  is$\mathrm{△}\ln s,$ which can be evaluated from (Template:EquationNote) as

Template:Big

Applying this and (Template:EquationNote) to the previous result gives

Template:Big

i.e., since Template:Mvar is the Template:Mvarth component ofTemplate:Math  and therefore ofTemplate:Math

Template:Big

i.e., since  Template:Math

Template:Big

Multiplying this equation by Template:Math  and adding it to equation (Template:EquationNote) gives

Template:Big

As we are proposing to integrate over a volume containing no sources, we can invoke the (homogeneous) wave equation (Template:EquationNote), so that the right-hand side is zero, leaving Template:NumBlk

Now at last we can substitute (Template:EquationNote) and (Template:EquationNote) into (Template:EquationNote) and get

$\begin{array}{cc}\underset{S}{\iint }& (\frac{1}{s}[{\partial }_n\psi ]-\frac{1}{cs}{\partial }_{n}s[\dot{\psi }]+\frac{1}{s^2}[\psi ]{\partial }_{n}s)dS\\ & =\underset{V}{\iiint }-\frac{2}{c}\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\mathrm{\nabla }s}{s}\right[\dot{\psi }\left]\right)dV+4\pi \underset{V}{\iiint }[\psi ]\delta (𝐫-{𝐫}^{\prime })dV,\end{array}$

in which we can rewrite the first volume integral as

$\begin{array}{cc}\underset{S}{\iint }-\frac{2}{c}\frac{\mathrm{\nabla }s}{s}\left[\dot{\psi }\right]\cdot \widehat{𝐧}dS& =\underset{S}{\iint }-\frac{2}{cs}\mathrm{\nabla }s\cdot \widehat{𝐧}\left[\dot{\psi }\right]dS\\ & =\underset{S}{\iint }-\frac{2}{cs}{\partial }_{n}s\left[\dot{\psi }\right]dS\end{array}$

and then transpose it to the left side, obtaining Template:NumBlk where the last term in the integrand on the left has been reverted to its original form (reversing the chain rule), and the remaining integral on the right isTemplate:Math  if position Template:Math is insideTemplate:Mvar, but zero if it is outside.

Although we derived (Template:EquationNote) by supposing, just after (Template:EquationNote), that Template:Mvar  is scalar, that restriction can now be dropped on account of the linearity of the various operators in (Template:EquationNote).

Although we derived (Template:EquationNote) by supposing that Template:Mvaris a finite  region, we can extend the result to an infinite region by adding another sheet to the bounding surfaceTemplate:Mvar  in such a way that (i) the region becomes finite, but (ii) the additional sheet makes no contribution to the surface integral. The simplest way to do this is to suppose that the additional sheet is at such a large distance that the disturbance has not reached it yet! By this expedient we can apply (Template:EquationNote) not only to the region inside a closed surface, but also (e.g.) to the region outside a closed surface, or the region on one side of an infinite open surface.

Although we derived (Template:EquationNote) by supposing, as usual in this paper, that Template:Math points out ofTemplate:Mvar  and that Template:Mvaris measured out ofTemplate:Mvar, this has the arguably counterintuitive implication that Template:Math is typically against the direction of propagation—directly against it in the simplest case, in which Template:Mvar  is the exterior of a sphere with a monopole source at its center.

So, in the following formal statement of our result, let us drop the symbol Template:Mvar  and define Template:Mvar  as being measured out of the region containing the sources, and consequently into the region that satisfies the homogeneous wave equation, changing the signs  in (Template:EquationNote).

Kirchhoff's integral theorem:  If

• the wave function Template:Mvar  satisfies the wave equation (with speedTemplate:Mvar) in a regionTemplate:Mvar  bounded by a surfaceTemplate:Mvar  (with all sources consequently on the other side ofTemplate:Mvar), and
• Template:Mvar  is the distance of the general point at positionTemplate:Math  from the observation point at positionTemplate:Math and
• quantities in square brackets are to be delayed byTemplate:Math and
• Template:Mvar  is the normal coordinate measured from the general point onTemplate:Mvar  intoTemplate:Mvar  [contrary to the usual direction for a named region, and contrary to the convention we have used above!],

then the expression Template:NumBlk is equal to the wave function atTemplate:Math  ifTemplate:Math  is insideTemplate:Math but zero if it is outside.

The above derivation (guided by Baker & Copson, 1939, pp. 38–40) is unusual in that it does not assume sinusoidal time-dependence at any stage—but objectionable in that (i) the quantity whose divergence we seek, after equation (Template:EquationNote), is a wild guess, and (ii) there are places where a term with coefficientTemplate:Math found by one path is merged with a term with coefficientTemplate:Math found by another path, as if the argument contained hidden redundancies that ought to be eliminated. A more traditional derivation (e.g., Baker & Copson, 1939, pp. 36–7; Born & Wolf, 2002, pp. 420–21) would first derive the special case for sinusoidal time-dependence (due to Helmholtz) from Green's identities, and then generalize the time-dependence. The traditional approach has the advantages of being less tortuous and more readily applicable to dispersive  media (in which Template:Mvaris frequency-dependent), but the disadvantages of depending on complex numbers and on the premise that a general function of time can be expressed as a sum of sinusoids. Helmholtz's integrand is a sinusoidal version of our expression (Template:EquationNote) below. One can derive that expression and thence the Kirchhoff integral in a far more elementary and accessible manner, albeit with some loss of rigor, by assuming (instead of justifying) the form of the wave function due to a monopole source (Putland, 2022). That approach first obtains the required distribution of secondary sources and then expresses the wave function as a surface integral—instead of inferring the secondary sources from the surface integral, as we are about to do here.

Monopole and dipole secondary sources

There is a convention whereby Template:Math denotes the derivative w.r.t.Template:Mvar  of the Template:Mvarth component of the vector fieldTemplate:Math so that the corresponding derivative of the entire vector is writtenTemplate:Math where the strange leading comma in the subscript announces that we want a partial derivative, not a component. Accordingly, let us write the derivative w.r.t.Template:Mvar  of the generic fieldTemplate:Mvar  asTemplate:Math.  And let it be understood that any "commaTemplate:Math" derivative is to be evaluated locally, and not as observed atTemplate:Math; that is, it does not  account for the variation of Template:Mvar  throughTemplate:Mvar.

In that notation, the integrand inside the big parentheses in (Template:EquationNote) can be written out in full as

$\begin{array}{cc}\psi & (t-s/c){\partial }_n\frac{1}{s}-\frac{1}{cs}{\partial }_{n}s{\psi }^{\prime }(t-s/c)-\frac{1}{s}{\psi }_{,n}(t-s/c)\\ & =\psi (t-s/c){\partial }_n\frac{1}{s}+\frac{1}{s}{\psi }^{\prime }(t-s/c)(-1/c){\partial }_{n}s-\frac{1}{s}{\psi }_{,n}(t-s/c)\\ & =\psi (t-s/c){\partial }_n\frac{1}{s}+\frac{1}{s}{\partial }_n\psi (t-s/c)-\frac{1}{s}{\psi }_{,n}(t-s/c),\end{array}$

i.e. Template:NumBlk where Template:Mvar  does  account for the variation of Template:Mvar  throughTemplate:Mvar. IfTemplate:Mvar is a small  change inTemplate:Math from  Template:Mvar  to  Template:Math the integrand can be written Template:NumBlk The second term (including the minus sign) is recognizable as the contribution to the wave function from a monopole source with strengthTemplate:Math. Similarly, in the first term, the expression in the big parentheses is the contribution from a monopole source with strengthTemplate:Math; and the operator Template:Mvar gives the change in that contribution due toTemplate:Mvar  increasing from Template:Mvar  toTemplate:Math  i.e. the change in that contribution due to moving the said monopole from  Template:Mvar  to  Template:Math  i.e. the whole contribution due to the combination of a monopole with strengthTemplate:Math  at  Template:Mvar  and a monopole with strengthTemplate:Math  at  Template:Math. This combination is called a dipole (or doublet) with strengthTemplate:Mvar  in the normal (Template:Mvar) direction.

According to (Template:EquationNote), the expression (Template:EquationNote) is to be scaled by Template:Math  and integrated over the surfaceTemplate:Mvar. Thus the secondary source distribution can be described as a monopole distribution of strength density Template:Math  plus a normal dipole distribution of strength densityTemplate:Math where "strength density" means strength per unit area. This description is well known.

The implication is not that the specified secondary sources really exist, or even that they could  exist, but only that the wave function in the regionTemplate:Mvar  is as if  it had been generated by the specified secondary sources (which would also give a null wave function outside the region). We should note, however, that a monopole contribution of the form (Template:EquationNote) can  really exist, even for a vector wave function, notwithstanding that it requires not only the magnitude but also the direction of the vector to be independent of the direction of propagation. That requirement might seem to exclude electromagnetic waves, for which the electric and magnetic fields are transverse to the direction of propagation and therefore not independent of it. But it is possible to describe such waves in terms of an electric scalar potential and a magnetic vector potential, such that the contribution to the latter from a current element has the same direction as the current element for all directions of propagation.

Generalized spatiotemporal-dipole (GSTD) secondary sources

The "dipole" discussed so far is a spatial  dipole, in which the constituent monopoles differ only in sign and by a small spatial displacement. If we add a small time shift between the monopoles, equal to the propagation time from one to the other, we get what David A.B. Miller (1991) calls a spatiotemporal  dipole. If we allow the time shift to be smaller than that propagation time, and allow one monopole to be attenuated by a small fraction, we get what I call a generalized  spatiotemporal dipole, the usefulness of which will now be demonstrated.

In the Helmholtz–Kirchhoff integrand (Template:EquationNote), the second term (including the sign) represents a monopole strength densityTemplate:Math  and the first term represents a dipole strength densityTemplate:Mvar  in the Template:Mvardirection; the dipole source per unit area ofTemplate:Mvar  is a spatial  dipole comprising a monopole with strengthTemplate:Math  at  Template:Mvar  (the inverted  monopole), and a monopole with strengthTemplate:Math  at  Template:Math (the uninverted  monopole), where Template:Mvar  is small (and the indicated strength densities are eventually to be divided byTemplate:Math). The idea is that by suitably modifying the spatial dipole, we might eliminate the need for the separate monopole and thus effectively reduce a tripole to a dipole. Let us try delaying the strength function of the inverted monopole byTemplate:Math and reducing its magnitude by a fractionTemplate:Mvar  (e.g., no reduction if Template:Math), where Template:Mvar andTemplate:Math are small  quantities to be determined. Then, compared with the uninverted monopole, the inverted monopole is recessed by the distanceTemplate:Math delayed by the timeTemplate:Math and attenuated by the fractionTemplate:Mvar.  At the field pointTemplate:Math according to (Template:EquationNote), the wave function due to the uninverted monopole is Template:NumBlk So the wave function due to the modified dipole is the total change in expression (Template:EquationNote) due to Template:Mvarincreasing byTemplate:Math  and Template:Mvarincreasing byTemplate:Math and the magnitude increasing by Template:Mvartimes its final value. Since Template:Mvarand Template:Mvar are small, that total change is

$h{\partial }_n\left(\frac{\psi (t-s/c)}{hs}\right)+{\tau }_h{\partial }_t\left(\frac{\psi (t-s/c)}{hs}\right)+{\alpha }_h\frac{\psi (t-s/c)}{hs},$

i.e.

${\partial }_n\left(\frac{\psi (t-s/c)}{s}\right)+\frac{{\tau }_h}{hs}\dot{\psi }(t-s/c)+\frac{{\alpha }_h}{hs}\psi (t-s/c),$

which will agree with (Template:EquationNote) if and only if, onTemplate:Math Template:NumBlk This is the sufficient and necessary condition for the modified dipoles to give the same secondary waves as the original dipoles and monopoles. And it does not look helpful in the general case, due to the independence of the wave function and its partial derivatives w.r.t.Template:Mvar  andTemplate:Mvar.

But now suppose that we have a single monopole primary source, so that the wave function Template:Mvar  is given by (Template:EquationNote). It is readily confirmed that this wave function and its partial derivatives are related by Template:NumBlk Applying the chain rule to the right side of (Template:EquationNote) gives

$\frac{{\tau }_h}{h}\dot{\psi }+\frac{{\alpha }_h}{h}\psi =-{\partial }_r\psi {\partial }_{n}r.$

So, substituting from (Template:EquationNote) and noting that Template:Mvar  is Template:Math  i.e. the cosine of the angle between Template:MathandTemplate:Math  we have Template:NumBlk To satisfy this for all Template:Math  andTemplate:Math we equate the coefficients ofTemplate:Math obtaining Template:NumBlk and equate the coefficients ofTemplate:Math obtaining Template:NumBlk so that the parameters of our "modified" dipole are uniquely determined. This "modified" dipole is what I have called a "generalized spatiotemporal dipole" (GSTD). The integrand in (Template:EquationNote) may then be understood as a distribution of GSTDs onTemplate:Math oriented normal toTemplate:Math the first term in the braces representing the spatial aspect (equal and opposite monopoles) and the second term (inTemplate:Math) representing the modifications (delay and attenuation of the inverted monopole). According to (Template:EquationNote), the delay of the inverted monopole is such that the waves from the two monopoles are synchronized (with opposing amplitudes) in the direction of the primary source, and in the cone of directions which make the same angle with the normalTemplate:Math as the primary source; this cone includes the direction of specular reflection offTemplate:Mvar. And according to (Template:EquationNote), the attenuation of the inverted monopole is such that the waves from the two monopoles cancel at a distanceTemplate:Mvar  in any of these directions (including at the primary source); at that distance, the greater proximity of the inverted monopole compensates for the reduced strength. Thus there are two ways in which the GSTDs suppress "backward" secondary waves: collectively, they produce a wave function which is null on the primary source's side of the surfaceTemplate:Mvar; individually, they suppress secondary waves at particular distances in particular directions, including the direction of specular reflection offTemplate:Mvar.

If Template:Mvar  coincides with a primary wavefront, so that its normalTemplate:Math is parallel toTemplate:Math then we have  Template:Math  in (Template:EquationNote), so that the delay Template:Mvar becomesTemplate:Math which is simply the time taken for the waves emitted by the uninverted monopole to reach the inverted monopole. The latter is in the Template:Mathdirection, which is therefore the direction in which the waves from the two monopoles are synchronized (and cancel at distanceTemplate:Mvar); the "cone of directions" collapses to its axis.

If the primary wavefront is plane, we have  Template:Math  in (Template:EquationNote), so that Template:Math; the inverted monopole is not attenuated, and the cancellation of the waves from the two monopoles (in the cone with its axis in theTemplate:Mathdirection) becomes a far-field effect.

So if Template:Mvar  coincides with a primary wavefront and  is plane, the inverted monopole is delayed byTemplate:Math  and unattenuated, so that the waves from the two monopoles cancel in the Template:Mathdirection in the far field. This case is what Miller (1991) calls a "spatiotemporal dipole". We have "generalized" it in two ways: by allowing the delay of the inverted monopole to be less thanTemplate:Math so that the direction of cancellation need not be normal toTemplate:Mvar; and by allowing the inverted monopole to be attenuated, so that the cancellation may occur at a finite distance. Together, these modifications allow the surface of integrationTemplate:Mvar  to be of a general shape and orientation and at a general distance from the primary source.

Although Miller applied his spatiotemporal-dipole theory to "uniform spherical or plane wave fronts" (Miller, 1991, after eq. 5), it is in fact a plain-wave (far-field) approximation in that it neglects the Template:Math decay in the magnitude of the primary wave, with the result that his equation (4), which corresponds to our (Template:EquationNote), lacks the second term on the right. To make the theory exact for allTemplate:Math we need the inverted monopoles to be attenuated in accordance with (Template:EquationNote).

Application to diffraction by an aperture

Suppose that the primary sources are partly obstructed by an opaque baffle with an aperture in it, and that we are interested in the wave function that propagates beyond the baffle. Let us choose a surfaceTemplate:Mvar  consisting of two segments, namely Template:Mvarspanning the aperture, and Template:Mvaron the side of the baffle facing away from the sources (the dark side or quiet side of the baffle). The obvious way to proceed is to suppose that the baffle simply eliminates the secondary sources onTemplate:Mvar while leaving the secondary sources onTemplate:Mvar unchanged (as if the baffle were not there). The result, as far as the wave function inTemplate:Mvar (beyond the baffle) is concerned, is simply that the integral in (Template:EquationNote) is taken over Template:Mvaronly.

Integrating over the aperture alone is indeed the standard answer, but there are various other ways of explaining it. Some explanations, including the famously inconsistent one offered by Kirchhoff himself, are discussed in Putland, 2022 (§ 2.2 and Appendices A & B), with references to other works.

Additional information

Acknowledgments

Professor Chen-To Tai, Template:Serif, died in 2004.  He first came to my attention in 2018 through his paper "On the presentation of Maxwell's theory" (Proc. Template:Serif, 60(8): 936–45, 1972). In nearly every place where I mention him here, even if I do not accept his conclusion, I am entirely indebted to his works for drawing my attention to the issue raised. In particular, it was he who alerted me to Gibbs's original definitions of the divergence and curl and their suitability for expression in indicial notation (Tai, 1995, pp. 17, 21). And although he might not have been pleased, it was through him that I first knew with certainty that the del-dot and del-cross notations work in general coordinates (ibid., pp. 64–5).

This article uses images from Wikimedia Commons.

The essence of the above derivation of "generalized" spatiotemporal-dipole secondary sources has been previously published (Putland, 2022, § 3.7), but not previously peer-reviewed.

Competing interests

None.

Ethics statement

This article does not concern research on human or animal subjects.

TO DO:

• More illustrations?

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Further reading

M.J. Crowe, "A History of Vector Analysis" (address at the University of Louisville, Autumn term, 2002), researchgate.net/publication/244957729_A_History_of_Vector_Analysis (including much discussion of quaternions).

P. Lynch, "Matthew O'Brien: An inventor of vector analysis", Bulletin of the Irish Mathematical Society, No. 74 (Winter 2014), pp. 81–8; doi.org/10.33232/BIMS.0074.81.88.