nLab
curl

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

In Riemannian geometry, the curl or rotation of a vector field XX over an oriented 33-dimensional Riemannian manifold (M,g)(M,g) is the vector field curl(X)curl(X) (or rot(X)rot(X)) defined by

curl(X)=g 1 gd dRg(X), curl(X) = g^{-1}\star_g d_{dR}g(X) ,

where g\star_g is the Hodge star operator of (M,g)(M,g),

g:Ω i(M;)Ω 3i(M;) \star_g\colon \Omega^i(M;\mathbb{R}) \to \Omega^{3-i}(M;\mathbb{R})

Alternatively, the curl/rotation of a vector field 𝒜\vec\mathcal{A} in some point xMx\in M is calculated (or alternatively defined) by the integral formula

nrot𝒜=lim areaS01areaS St𝒜dr \vec{n}\cdot rot \vec\mathcal{A} = \lim_{area S\to 0} \frac{1}{area S} \oint_{\partial S} \vec{t}\cdot \vec\mathcal{A} d r

where DD runs over the smooth (pseudo)-oriented surfaces (smooth submanifolds of dimension 22) containing the point xx and with smooth boundary D\partial D, n\vec{n} is the unit vector of outer normal to the surface SS, and t\vec{t} is the unit vector tangent to the curve S\partial S. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and balls with decreasing radius in this particular coordinate chart.

(We use the orientation of MM in the Hodge dual, or alternatively in determining the direction of n\vec{n} from the orientation of SS or the direction of t\vec{t} fom the pseudo-orientation of SS.)

More generally, if (M,g)(M,g) is a Riemannian manifold whose cotangent spaces (equivalently, tangent spaces) are smoothly equipped with a binary cross product :Ω 2(M;R)Ω 1(M;R)⨉\colon \Omega^2(M;R) \to \Omega^1(M;R), then the curl of any vector field XX is

curl(X)=g 1d dRg(X). curl(X) = g^{-1} ⨉ d_{dR} g(X) .

However, this is not as general as it may appear:

  • in 00 or 11 dimension, the cross product, hence the curl, must always be 00;
  • in 33 dimensions, a smooth choice of cross product is equivalent to a smooth choice of orientation, and we recover the previous formula;
  • in 77 dimensions, if a smooth choice of cross product is possible (as on the 77-sphere), then uncountably many are possible, giving as many different notions of curl;
  • in any other number of dimensions, no binary cross product exists at all, hence no curl.

There are also cross products of other arity? in other dimensions; using essentially the same formula, we can take the curl of a kk-vector field? if we have a smooth (k+1)(k+1)-ary cross product.

Examples

If (M,g)(M,g) is 3\mathbb{R}^3 endowed with the canonical Euclidean metric, then the curl of a vector field (X 1,X 2,X 3)=X 1 1+X 2 2+X 3 3(X^1,X^2,X^3) = X^1\partial_1 + X^2\partial_2 + X^3\partial_3 is

curl(X) 1=X 3x 2X 2x 3;curl(X) 2=X 1x 3X 3x 1;curl(X) 3=X 2x 1X 1x 2 curl(X)^1 = \frac{\partial X^3}{\partial x^2}-\frac{\partial X^2}{\partial x^3} ;\qquad curl(X)^2 = \frac{\partial X^1}{\partial x^3}-\frac{\partial X^3}{\partial x^1} ;\qquad curl(X)^3 = \frac{\partial X^2}{\partial x^1}-\frac{\partial X^1}{\partial x^2}

This is the classical curl from vector analysis?.

Remark

In many classical applications of the curl in vector analysis?, the Riemannian structure is actually irrelevant, and the gradient can be replaced with the deRham differential d dRd_{dR}. That is, XX is treated as the 11-form g(X)g(X), its curl is treated as the 22-form d dRg(X)d_{dR} g(X), and once these identifications are made there is no need to involve gg at all.

Revised on June 11, 2013 02:09:39 by Toby Bartels (64.89.53.249)