alternate characterization of curl

Let 𝐅 be a smooth vector fieldMathworldPlanetmath on (an open subset of) 3.

We show that curl𝐅 defined using the coordinate-free definition given on the parent entry ( is the same as the curl defined by ×𝐅 in Cartesian coordinatesMathworldPlanetmath.

The case for spherical surfaces

This will be done by directly computing the limit 𝐋 of surface integrals defining curl𝐅(𝐩), using spheres S2(r,𝐩) centered at 𝐩 of radius r. The formulaMathworldPlanetmathPlanetmath is:

curl𝐅(𝐩)=𝐋 =limr034πr3S2(r,𝐩)𝐧×𝐅𝑑A

where 𝐧 is the outward unit normal to the surface (at each point of the surface), and S2 is the unit sphereMathworldPlanetmath at the origin.

We simplify the last integral. Expanding 𝐅(r𝐧+𝐩) in a first-degree Taylor polynomial about 𝐩, we have

S2𝐧×𝐅(r𝐧+𝐩)𝑑A =S2𝐧×𝐅(𝐩)𝑑A

The integral S2𝐧×𝐅(𝐩)𝑑A vanishes by symmetryMathworldPlanetmathPlanetmath of the sphere, while


Combining these facts, we obtain

𝐋 =limr0[0+34πrS2𝐧×D𝐅(𝐩)r𝐧𝑑A+o(1)]

Notice that 𝐋 depends only on the derivativePlanetmathPlanetmath of 𝐅 at 𝐩.

We want to evaluate the last integral in Cartesian coordinates. Let 𝐞k be an orthonormal basisMathworldPlanetmath of 3 oriented positively, and let B be the matrix of the derivative D𝐅(p) in this basis. Then the kth coordinateMathworldPlanetmathPlanetmath of 𝐋 with respect to the same basis is


The kth coordinate of the integrand is


where to lessen the writing, we employ the Einstein summation convention, along with the Levi-Civita permutation symbol ϵijk, and Blj denotes the entry at the jth row, lth column of B.

In the summation above, if a summmand has il, then the integral of that summand over the sphere is zero, by symmetry. This means that in the summation the index l may be set to i, and thus


Now there is a formula for the evaluation of integrals of polynomials over Sm-1m, in terms of the gamma function; in our case (m=3) the formula reads:


(If you do not know this formula, the integral in our case can be computed directly using spherical coordinatesMathworldPlanetmath.) Therefore the kth component of 𝐋 is


But this is just (×𝐅(𝐩))𝐞k.

The case for arbitrary surfaces

Although we have only computed


only for spheres S=S2(r,𝐩), this formula holds for arbitrary closed surfaces S that shrink nicely to 𝐩. It is hardly obvious, especially since our computation before depended on the symmetry of the sphere extensively.

To show the general result, consider the triple scalar product (𝐯×𝐅)𝐞k. This is a linear functionalMathworldPlanetmathPlanetmath in the vector 𝐯, so there exists a unique vector function 𝐠k such that (𝐯×𝐅)𝐞k=𝐠k𝐯 for all 𝐯3. We can find the components of this 𝐠k by evaluating the functionalMathworldPlanetmathPlanetmath at 𝐯=𝐞i:


The reason for considering such expressions is that, putting 𝐯=𝐧, we have


So we have converted the original integral into an ordinary surface integral. And this surface integral can be changed into a volume integral, by using the divergence theoremMathworldPlanetmathPlanetmath:


where M is the volume whose boundary is S. Hence

𝐋𝐞k =limV01VS(𝐧×𝐅)𝐞k𝑑A

Definition in terms of differential forms

We mention, in passing, a computational, yet coordinate-free, alternative to the definition of the curl, using differential formsMathworldPlanetmath. If ω is a 1-form on 3 such that ω(𝐯)=𝐅,𝐯, then the curl of 𝐅 is defined as the vector function 𝐠=gkek such that


In Cartesian coordinates, we have

ω =F1dx1+F2dx2+F3dx3
dω =g1dx2dx3+g2dx3dx1+g3dx1dx2,

If we take the exterior derivative of the first equation for ω, and then equate components with the second equation for dω, we find that gk = (×𝐅)𝐞k, so our new definition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the others.

Title alternate characterization of curl
Canonical name AlternateCharacterizationOfCurl
Date of creation 2013-03-22 15:29:10
Last modified on 2013-03-22 15:29:10
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 9
Author stevecheng (10074)
Entry type DerivationMathworldPlanetmath
Classification msc 53-01
Related topic curl
Related topic nabla
Related topic FirstOrderOperatorsInRiemannianGeometry
Related topic Curl
Related topic NablaNabla