# nLab curl

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

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.

## Examples

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?.

## 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 $d_{dR}$. 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)