curl

In Riemannian geometry, the **curl** or **rotation** of a vector field $X$ over an oriented $3$-dimensional Riemannian manifold $(M,g)$ is the vector field $curl(X)$ (or $rot(X)$) defined by

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

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

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

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

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

where $D$ runs over the smooth (pseudo)-oriented surfaces (smooth submanifolds of dimension $2$) containing the point $x$ and with smooth boundary $\partial D$, $\vec{n}$ is the unit vector of outer normal to the surface $S$, and $\vec{t}$ is the unit vector tangent to the curve $\partial S$. 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 $M$ in the Hodge dual, or alternatively in determining the direction of $\vec{n}$ from the orientation of $S$ or the direction of $\vec{t}$ fom the pseudo-orientation of $S$.)

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

$curl(X) = g^{-1} ⨉ d_{dR} g(X)
.$

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

- in $0$ or $1$ dimension, the cross product, hence the curl, must always be $0$;
- in $3$ dimensions, a smooth choice of cross product is equivalent to a smooth choice of orientation, and we recover the previous formula;
- in $7$ dimensions, if a smooth choice of cross product is possible (as on the $7$-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 $k$-vector field? if we have a smooth $(k+1)$-ary cross product.

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

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

This is the classical curl from vector analysis?.

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 $d_{dR}$. That is, $X$ is treated as the $1$-form $g(X)$, its curl is treated as the $2$-form $d_{dR} g(X)$, and once these identifications are made there is no need to involve $g$ at all.

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