PlanetPhysics/Curl

To the [[../VectorProduct/|cross product]] of the [[../Gradient/|gradient operator]] $\mathrm{\nabla }\times$ Maxwell gave the name $𝐜url$.

$\mathrm{\nabla }\times 𝐕=curl𝐕$

The curl of a [[../VectorFunctions/|vector function]] $𝐕$ is itself a vector function of [[../Position/|position]] in space. As the name indicates, it is closely connected with the angular [[../Velocity/|velocity]] or [[../QuarkAntiquarkPair/|spin]] of the [[../AbsoluteMagnitude/|flux]] at each point. But the interpretation of the curl is neither so easily obtained nor so simple as that of the [[../DivergenceOfAVectorField/|divergence]].

Consider as before that $𝐕$ represents the flux of a fluid. Take at a definite instant an infinitesimal sphere about any point $(x,y,z)$. At the next instant what has become of the sphere? In the first place it may have moved off as a whole in a certain direction by an amount $d𝐫$. In other words it may have a translational velocity of $d𝐫/dt$. In other words it may have undergone such a [[../CohomologicalProperties/|deformation]] that it is no longer a sphere. It may have been subjected to a strain by virtue of which it becomes slightly ellipsoidal in shape. Finally it may have been rotated as a whole about some axis through an angle $dw$. That is to say, it may have an angular velocity the [[../AbsoluteMagnitude/|magnitude]] of which is $dw/dt$. An infinitesimal sphere therefore may have any one of these distinct [[../Bijective/|types]] of [[../CosmologicalConstant/|motion]] or all of them combined. First, a translation with definite velocity. Second, a strain with three definite rates of elongation along the axes of an ellipsoid. Third, an angular velocity about a difinite axis. It is this third type of motion which is given by the curl. In fact, the curl of the flux $V$ is a [[../Vectors/|vector]] which has at each point of space the direction of the instantaneous axis of rotation at that point and a magnitude equal to twice the instantaneous angular velocity about that axis.

The analytic discussion of the motion of a fluid presents more difficulties than it is necessary to introduce in treating the curl. The motion of a [[../CenterOfGravity/|rigid body]] is sufficiently complex to give an adequate idea of the [[../Cod/|operation]]. It was seen that the velocity of the [[../Particle/|particles]] of a rigid body at any instant is given by the [[../Formula/|formula]] $𝐯={𝐯}_0+𝐚\times 𝐫$

$curl𝐯=\mathrm{\nabla }\times 𝐯=\mathrm{\nabla }\times {𝐯}_0+\mathrm{\nabla }\times (𝐚\times 𝐫)$

Let

$𝐚=a_1\widehat{𝐢}+a_2\widehat{𝐣}+a_3\widehat{𝐤}$

$𝐫=r_1\widehat{𝐢}+r_2\widehat{𝐣}+r_3\widehat{𝐤}=x\widehat{𝐢}+y\widehat{𝐣}+z\widehat{𝐤}$

expand $\mathrm{\nabla }\times (𝐚\times 𝐫)$ formally as if it were the [[../BACKCAB/|Vector Triple Product]] of $\mathrm{\nabla }$, $𝐚$, and $𝐫$. Then

$\mathrm{\nabla }\times 𝐯=\mathrm{\nabla }\times {𝐯}_0+(\mathrm{\nabla }\cdot 𝐫)𝐚-(\mathrm{\nabla }\cdot 𝐚)𝐫$

${𝐯}_0$ is a constant vector. Hence the term $\mathrm{\nabla }\times {𝐯}_0$ vanishes.

$\mathrm{\nabla }\cdot 𝐫=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=3$

Since $𝐚$ is a constant vector, it may be placed upon the other side of the differential [[../QuantumSpinNetworkFunctor2/|operator]], $\mathrm{\nabla }\cdot 𝐚=𝐚\cdot \mathrm{\nabla }$

$𝐚\cdot \mathrm{\nabla }𝐫=(a_1\frac{\partial }{\partial x}+a_2\frac{\partial }{\partial y}+a_3\frac{\partial }{\partial z})𝐫=a_1\widehat{𝐢}+a_2\widehat{𝐣}+a_3\widehat{𝐤}=𝐚$

Hence

$\mathrm{\nabla }\times 𝐯=3𝐚-𝐚=2𝐚$

Therefore in the case of the motion of a rigid body the curl of the linear velocity at any point is equal to twice the angular velocity in magnitude and in direction.

$\mathrm{\nabla }\times 𝐯=curl𝐯=2𝐚$ $𝐚=\frac{1}{2}\mathrm{\nabla }\times 𝐯=\frac{1}{2}curl𝐯$ $𝐯={𝐯}_0+\frac{1}{2}(\mathrm{\nabla }\times 𝐯)\times 𝐫={𝐯}_0+\frac{1}{2}\left(curl𝐯\right)\times 𝐫$

The expansion of $\mathrm{\nabla }\times (𝐚\times 𝐫)$ formally may be avoided by multiplying $𝐚\times 𝐫$ out and then applying the operator $\mathrm{\nabla }\times$ to the result.

References

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

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