nLab curl

Definitions

In terms of differential forms

In Riemannian geometry, the curl or rotation of a vector field $v$ on an oriented $3$-dimensional Riemannian manifold $(X,g)$ is the vector field denoted $curl(v)$ (or $rot(v)$ or $\Del \times v$) defined by

$$curl(v) \;\coloneqq\; g^{-1} \left(\star_g d_{dR}\, g(v) \right) \,.$$

where

1. $\Gamma(T X) \underoverset{\underset{g}{\longrightarrow}}{\overset{g^{-1}}{\longleftarrow}}{\phantom{AA}\simeq\phantom{AA}} \Omega^1(X)$ is the linear isomorphism between vector fields and differential 1-forms given by the metric tensor $g$;

2. $d_{dR} \;\colon\; \Omega^n(X) \longrightarrow \Omega^{n+1}(X)$ is the de Rham differential

3. $\star_g \;\colon\; \Omega^n(X) \to \Omega^{dim(X)-n}(X)$ is the Hodge star operator (which uses the orientation of $X$).

Notice that for this to make sense it is crucial that the dimension of $X$ is $3$, for only then is the Hodge dual of the de Rham differential of a $1$-form again a $1$-form; that is, $n = 3$ is the unique solution of $n - (1 + 1) = 1$.

Via integration

Alternatively, the curl/rotation of a vector field $\vec v$ at some point $x\in X$ may be defined as the limit integral formula

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

where $D$ runs over the smooth oriented surfaces (submanifolds of dimension $2$) containing the point $x$ and with smooth boundary $\partial D$, $\vec{n}$ is the unit vector through the surface $S$, and $\vec{t}$ is the unit vector tangent to the curve $\partial S$. (We use the orientation of $X$ to determine the direction of $\vec{n}$ from the orientation of $S$.)

This formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and discs with decreasing radius in this particular coordinate chart. One can even use $\pi r^2$ in place of the actual area of the disc around $x$ with coordinate-radius $r$, to save on calculating this area, as long as the coordinate chart assigns the standard coordinates to the metric at $x$.

The proof that this definition is coherent and agrees with the previous one is essentially the Kelvin–Stokes Theorem; see below for discussion.

Via cross products

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

$$curl(v) \;=\; g^{-1} ⨉ d_{dR} g(v)$$

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.

That said, there are also cross products of other arity? in other dimensions; using essentially the same formula, we can take the curl of a $k$-multivector field if we have a smooth $(k+1)$-ary cross product. Or if the cross product is other than vector-valued, then we can obtain a curl that is other than a vector field.

In particular, in $2$ dimensions, we have the scalar curl

$$curl(v) \;=\; ⨉ d_{dR} g(v)$$

where $⨉\colon \Omega^2(X;\mathbb{R}) \to \Omega^0(X;\mathbb{R})$ is the volume form (or area form) on the $2$-dimensional Riemannian manifold $X$.

Examples

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

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

This is the classical curl from vector analysis?.

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

$$curl(v) = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2} .$$

Relation to the Stokes theorems

Recall that if $X$ is an $n$-dimensional differentiable manifold, $D$ is a $p$-dimensional submanifold with boundary, and $\alpha$ is a differentiable $(p-1)$-rank exterior differential form on a neighbourhood of $D$ in $X$, then the generalized Stokes Theorem says that the integral of $\alpha$ on the boundary $\partial{D}$ equals the integral on $S$ of the de Rham differential $\mathrm{d}_{DR}\alpha$.

When $n=3$, $p=2$, and $X$ is equipped with an orientation and a metric, then this is equivalent to saying that the integral of a vector field $v$ along the boundary of an oriented surface $D$ in $X$ is equal to the integral of the vector field's curl across the surface:

(1)$$\int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \cdot \mathrm{d}\mathbf{S} .$$

(In particular, when $X$ is $\mathbb{R}^3$ with its standard orientation and metric, then this is equivalent to the classical Kelvin–Stokes Theorem.) At least, this is what it says if the curl is defined in terms of differential forms; if the curl is defined via integration instead, then (1) is immediate, and the Kelvin–Stokes Theorem says that this definition matches the other one.

When $n=2$, $p=2$, and $X$ is equipped with an orientation and a metric, then the Stokes Theorem is equivalent to saying that the integral of a vector field $v$ along a simple closed curve in $X$ is equal to the integral of the vector field's scalar curl on the region $D$ enclosed by the curve:

(2)$$\int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \mathrm{d}A .$$

(In particular, when $X$ is $\mathbb{R}^2$ with its standard orientation and metric, then this is equivalent to the curl-circulation form of Green's Theorem.)

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 $g$ or $X$ directly at all.

Related concepts

• nabla

• gradient

  • gradient flow

  • symplectic gradient

• Hamiltonian flow

• curl

• divergence

References

See also

• Wikipedia, Curl