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alternate characterization of curl
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(Derivation)
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Let $\vF$ be a smooth vector field on (an open subset of) $\real^3$ .
We show that $\curl \vF$ defined using the coordinate-free definition given on the parent entry is the same as the curl defined by $\nabla \cross \vF$ in Cartesian coordinates.
This will be done by directly computing the limit $\vL$ of surface integrals defining $\curl \vF(\vp)$ , using spheres $S^2(r, \vp)$ centered at $\vp$ of radius $r$ . The formula is:
where $\vn$ is the outward unit normal to the surface (at each point of the surface), and $S^2$ is the unit sphere at the origin.
We simplify the last integral. Expanding $\vF(r\vn + \vp)$ in a first-degree Taylor polynomial about $\vp$ , we have
The integral $\iint_{S^2} \vn \cross \vF(\vp) \, dA$ vanishes by symmetry of the sphere, while
Combining these facts, we obtain
Notice that $\vL$ depends only on the derivative of $\vF$ at $\vp$ .
We want to evaluate the last integral in Cartesian coordinates. Let $\ve_k$ be an orthonormal basis of $\real^3$ oriented positively, and let $B$ be the matrix of the derivative $\DF(p)$ in this basis. Then the $k$ th coordinate of $\vL$ with respect to the same basis is $$ \left(\iint_{S^2} \vn \cross B\vn \, dA \right)
\cdot \ve_k = \iint_{S^2} (\vn \cross B\vn) \cdot \ve_k \, dA $$ The $k$ th coordinate of the integrand is $$ (\vn \cross B\vn) \cdot \ve_k = n^i \, (B\vn)^j \, \epsilon_{ijk} = n^i \, B^j_l n^l \, \epsilon_{ijk}\,, $$ where to lessen the writing, we employ the Einstein summation convention, along with the Levi-Civita permutation symbol $\epsilon_{ijk}$ , and $B^j_l$ denotes the entry at the $j$ th row, $l$ th column of $B$ .
In the summation above, if a summmand has $i \neq l$ , 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 $$ \left(\iint_{S^2} \vn \cross B\vn \, dA \right) \cdot \ve_k = \iint_{S^2} n^i B_i^j n^i \epsilon_{ijk} \, dA = B_i^j \epsilon_{ijk} \iint_{S^2} (n^i)^2 \, dA\,. $$ Now there is a formula for the evaluation of integrals of polynomials over $S^{m-1} \subset \real^m$ , in terms of the gamma function; in our case ($m = 3$ ) the formula reads: $$ \iint_{S^2} (n^i)^2 \, dA = \frac{2 \Gamma(\frac{3}{2}) \, \Gamma(\frac{1}{2}) \, \Gamma(\frac{1}{2})}{\Gamma(\frac{3}{2} + \frac{1}{2} + \frac{1}{2})} = \frac{2 \Gamma(\frac{3}{2}) \, \sqrt{\pi} \, \sqrt{\pi}}{\frac{3}{2} \Gamma(\frac{3}{2})} = \frac{4\pi}{3}\,. $$ (If you do not know this formula, the integral in our case can be computed directly using spherical coordinates.) Therefore the $k$ th component of $\vL$ is $$ \vL \cdot \ve_k = \frac{3}{4\pi} \, B_i^j
\epsilon_{ijk} \iint_{S^2} (n^i)^2 \, dA = B_i^j \epsilon_{ijk} = \left.\frac{\partial F^j}{\partial x^i}\right|_\vp \, \epsilon_{ijk}\,. $$ But this is just $(\nabla \cross \vF(\vp)) \cdot \ve_k$ .
Although we have only computed $$ \vL = \curl \vF(\vp) = \lim_{V \to 0} \frac{1}{V} \iint_S \vn \cross \vF \, dA $$ only for spheres $S = S^2(r, \vp)$ , this formula holds for arbitrary closed surfaces $S$ that shrink nicely to $\vp$ . 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 $(\vv \times \vF) \cdot \ve_k$ . This is a linear functional in the vector $\vv$ , so there exists a unique vector function $\vg_k$ such that $(\vv \times \vF) \cdot \ve_k = \vg_k \cdot \vv$ for all $\vv \in \real^3$ . We can find the components of this $\vg_k$ by evaluating the
functional at $\vv = \ve_i$ : $$ g_k^i = \vg_k \cdot \ve_i = (\ve_i \times \vF) \cdot \ve_k = \det(\ve_i, \vF, \ve_k) = F^j \epsilon_{ijk}\,. $$ The reason for considering such expressions is that, putting $\vv = \vn$ , we have $$ \iint_{S} (\vn \times \vF) \cdot \ve_k \, dA = \iint_{S} \vg_k \cdot \vn \, dA = \iint_{S} \vg_k \cdot d\vA\,. $$ 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 theorem: $$ \iint_S \vg_k \cdot d\vA = \iiint_M \diverg \vg_k \, dV = \iiint_M \frac{\partial F^j}{\partial x^i} \, \epsilon_{ijk} \, dV\,, $$ where $M$ is the volume whose boundary is $S$ . Hence
We mention, in passing, a computational, yet coordinate-free, alternative to the definition of the curl, using differential forms. If $\omega$ is a 1-form on $\real^3$ such that $\omega(\vv) = \langle \vF, \vv \rangle$ , then the curl of $\vF$ is defined as the vector function $\vg = g^k \, e_k$ such that $$ d\omega(\vu, \vv) = \langle \vg, \vu \cross \vv \rangle\,. $$ In Cartesian coordinates, we have
If we take the exterior derivative of the first equation for $\omega$ , and then equate components with the second equation for $d\omega$ , we find that $g^k$ = $(\nabla \cross \vF) \cdot \ve_k$ , so our new definition is equivalent to the others.
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Cross-references: equivalent, equate, equation, exterior derivative, 1-form, differential forms, boundary, divergence theorem, volume, expressions, functional, components, function, vector, linear functional, triple scalar product, obvious, closed, component, spherical coordinates, gamma function, terms, polynomials, index, summation, column, row, Levi-Civita permutation symbol, Einstein summation convention, integrand, coordinate, basis, matrix, oriented, orthonormal basis, derivative, symmetry, vanishes, Taylor polynomial, origin, unit sphere, point, unit normal, formula, radius, spheres, integrals, surface, limit, Cartesian coordinates, curl, open subset, vector field, smooth
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This is version 6 of alternate characterization of curl, born on 2005-08-24, modified 2006-10-01.
Object id is 7341, canonical name is AlternateCharacterizationOfCurl.
Accessed 2298 times total.
Classification:
| AMS MSC: | 53-01 (Differential geometry :: Instructional exposition ) |
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Pending Errata and Addenda
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![$\displaystyle = \lim_{r \to 0} \left[ 0 + \frac{3}{4\pi r} \iint_{S^2} \mathbf{n}\times {\mathrm{D}}\mathbf{F}(\mathbf{p}) r\mathbf{n}\, dA + o(1) \right]$](http://images.planetmath.org:8080/cache/objects/7341/js/img9.png "$\displaystyle = \lim_{r \to 0} \left[ 0 + \frac{3}{4\pi r} \iint_{S^2} \mathbf{n}\times {\mathrm{D}}\mathbf{F}(\mathbf{p}) r\mathbf{n}\, dA + o(1) \right]$")
