PlanetPhysics/Gradient

The gradient is the [[../Vectors/|vector]] sum of the resultant rate of increase of a [[../Vectors/|scalar]] funcion $V$ and is denoted $\mathrm{\nabla }V$. It represents a directed rate of change of $V$. A directed derivative or vector derivative of $V$, so to speak. In cartesian coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial x}\widehat{𝐢}+\frac{\partial V}{\partial y}\widehat{𝐣}+\frac{\partial V}{\partial z}\widehat{𝐤}$

It is common to regard $\mathrm{\nabla }$ as the gradient operator which obtains a vector $\mathrm{\nabla }V$ from a scalar [[../Bijective/|function]] $V$ of [[../Position/|position]] in space.

$\mathrm{\nabla }V=(\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐢𝐤})V$

Thus it is easy to [[../Work/|work]] with just the gradient operator

$\mathrm{\nabla }=\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐤}$

This symbolic [[../QuantumSpinNetworkFunctor2/|operator]] $\mathrm{\nabla }$ was introduced by Sir W. R. Hamilton. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated [[../Formula/|formulas]] in which $\mathrm{\nabla }$ occurs a number of times no inconvenience to the speaker or hearer arises from the repetition. $\mathrm{\nabla }V$ is read simply as "del $V$."

Coordinate System Independence

Although this operator $\mathrm{\nabla }$ has been defined as

$\mathrm{\nabla }V=\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐤}$

so that it appears to depend upon the choice of the axes, it is in reality independent of them. This would be surmised from the interpretation of $\mathrm{\nabla }$ as the [[../AbsoluteMagnitude/|magnitude]] and direction of the most rapid increase of $V$. To demonstrate the independence take another set of axes, ${\widehat{𝐢}}^{\prime }$, ${\widehat{𝐣}}^{\prime }$, ${\widehat{𝐤}}^{\prime }$ and a new set of variables $x^{\prime }$, $y^{\prime }$, $z^{\prime }$ referred to them. Then $\mathrm{\nabla }$ referred to this [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] is

${\mathrm{\nabla }}^{\prime }=\frac{\partial }{\partial x^{\prime }}\widehat{{𝐢}^{\prime }}+\frac{\partial }{\partial y^{\prime }}\widehat{{𝐣}^{\prime }}+\frac{\partial }{\partial z^{\prime }}\widehat{{𝐤}^{\prime }}$

(Please Insert PROOF here...)

Leaving behind the proof of coordinate system independence, here is the gradient opertor in the most common coordinate systems.

Cartesian Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial x}\widehat{𝐢}+\frac{\partial V}{\partial y}\widehat{𝐣}+\frac{\partial V}{\partial z}\widehat{𝐤}$

Cylindrical Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial V}{\partial \theta }\hat{\mathit{\theta }}+\frac{\partial V}{\partial z}\widehat{𝐳}$

Spherical Coordinates

$\mathrm{\nabla }V=\frac{\partial V}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial V}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{1}{r\sin \varphi }\frac{\partial V}{\partial \theta }\hat{\mathit{\theta }}$

References

[1] Wilson, E. "Vector Analysis." Yale University Press, New Haven, 1913.

This entry is a derivative of the Public [[../Bijective/|domain]] work [1].