Curl

The curl ∇×F\nabla\times\mathbf{F} of a differentiable vector function F\mathbf{F} is a vector field that represents the circulation of F\mathbf{F} around a point. In simpler terms, it describes how much F\mathbf{F} curls or wraps around a point, like a whirlpool.

By definition, the curl is the limit of circulation over the boundary γ\gamma of an infinitesimal area dSdS:

∇×F=lim⁡dS→01dS∮γF⋅dr\nabla\times\mathbf{F}=\lim_{ dS \to 0 } \frac{1}{dS}\oint_{\gamma}\mathbf{F}\cdot d\mathbf{r}

It is more generally linked to the circulation by the curl theorem.

In Cartesian coordinates it can written as

∇×F=∣x^y^z^∂∂x∂∂y∂∂zFxFyFz∣=(∂Fz∂y−∂Fy∂z)x^+(∂Fx∂z−∂Fz∂x)y^+(∂Fy∂x−∂Fx∂y)z^\nabla\times\mathbf{F}=\begin{vmatrix} \mathbf{\hat{x}} & \mathbf{\hat{y}} & \mathbf{\hat{z}} \\ \frac{ \partial }{ \partial x } & \frac{ \partial }{ \partial y } & \frac{ \partial }{ \partial z } \\ F_{x} & F_{y} & F_{z} \end{vmatrix}=\left( \frac{ \partial F_{z} }{ \partial y } - \frac{ \partial F_{y} }{ \partial z } \right)\mathbf{\hat{x}}+\left( \frac{ \partial F_{x} }{ \partial z } -\frac{ \partial F_{z} }{ \partial x } \right)\mathbf{\hat{y}}+\left( \frac{ \partial F_{y} }{ \partial x } -\frac{ \partial F_{x} }{ \partial y } \right)\mathbf{\hat{z}}

Useful results

Given a generic position vector r\mathbf{r}, the following useful results hold:

∇×r=0,∇×(rrn)=0,∇×(rf(r))=0\nabla\times\mathbf{r}=0,\qquad \nabla\times\left( \frac{\mathbf{r}}{r^{n}} \right)=0,\qquad \nabla\times(\mathbf{r}f(r))=0

for any f(r)f(\mathbf{r}).

In curvilinear coordinates

To find the curvilinear form of the curl, we need to evaluate the circulation

∮γF⋅dr\oint_{\gamma}\mathbf{F}\cdot d\mathbf{r}

where γ\gamma is the infinitesimal curve loop generated by starting at (u,v,w)(u,v,w) and successively increasing uu and vv by infinitesimal amounts, holding ww constant. In the infinitesimal limit, this forms a surface that can be seen as a rectangle of length dlu=f dudl_{u}=f\ du, width dlv=g dvdl_{v}=g\ dv and area da=(fg) dudv w^d\mathbf{a}=(fg)\ dudv\ \mathbf{\hat{w}}, where ff, gg and hh are functions of position that are specific to a given coordinate system (see Curvilinear coordinates functions).

If we chose the coordinate system to be right-handed, the integral runs counterclockwise. The "bottom" side yields

dr=f du u^⇒F⋅dr=(fFu) dud\mathbf{r}=f\ du\ \mathbf{\hat{u}}\quad\Rightarrow \quad \mathbf{F}\cdot d\mathbf{r}=(fF_{u})\ du

The "top" side is the same, but with a flipped sign and fFufF_{u} evaluated at v+dvv+dv instead of vv. Together, these give

[−(fFu)∣v+dv+(fFu)∣v] du=−[∂∂v(fFu)] dudv[-(fF_{u})|_{v+dv}+(fF_{u})|_{v}]\ du=-\left[ \frac{ \partial }{ \partial v } (fF_{u}) \right]\ dudv

For the same reasons, the "left" and "right" sides give

[∂∂u(gFv)] dudv\left[ \frac{ \partial }{ \partial u } (gF_{v}) \right]\ dudv

so the total is

∮γF⋅dr=[∂∂u(gFv)−∂∂v(fFu)] dudv=1fg[∂∂u(gFv)−∂∂v(fFu)]w^⋅da\oint_{\gamma}\mathbf{F}\cdot d\mathbf{r}=\left[\frac{ \partial }{ \partial u } (gF_{v}) -\frac{ \partial }{ \partial v } (fF_{u}) \right]\ dudv=\frac{1}{fg}\left[ \frac{ \partial }{ \partial u } (gF_{v})-\frac{ \partial }{ \partial v } (fF_{u}) \right]\mathbf{\hat{w}}\cdot d\mathbf{a}

The coefficient of dad\mathbf{a} defines the ww component of the curl. The components in uu and vv can be composed in the same, with a cyclical permutation through all the coordinates. We get

∇×F≡1gh[∂∂v(hFw)−∂∂w(gFv)]u^+ 1fh[∂∂w(fFu)−∂∂u(hFw)]v^+ 1fg[∂∂u(gFv)−∂∂v(fFu)]w^\begin{align} \nabla\times\mathbf{F}\equiv &\frac{1}{gh}\left[ \frac{ \partial }{ \partial v } (hF_{w})-\frac{ \partial }{ \partial w } (gF_{v}) \right]\mathbf{\hat{u}} \\ +&\ \frac{1}{fh}\left[ \frac{ \partial }{ \partial w } (fF_{u})-\frac{ \partial }{ \partial u } (hF_{w}) \right]\mathbf{\hat{v}} \\ +&\ \frac{1}{fg}\left[ \frac{ \partial }{ \partial u } (gF_{v})-\frac{ \partial }{ \partial v } (fF_{u}) \right]\mathbf{\hat{w}} \end{align}

which gives

∮γF⋅dr=(∇×F)⋅da\oint_{\gamma}\mathbf{F}\cdot d\mathbf{r}=(\nabla\times\mathbf{F})\cdot d\mathbf{a}

This is almost Stokes' theorem, but it currently only applies to an infinitesimal loop. However, any (finite) surface can be broken down into many such loops, with edges overlapping on each other. Since the direction of integration is fixed by the basis, all integrations are done with the same direction, which means that overlapping sides will be integrated over in opposite directions. Since the sides are the same length, they cancel each other out. The only ones left are the external sides that are not adjacent to any other loop, which when combined form the exterior bounding loop of the shape. This works for any finite surface, so we can state

∮γF⋅dr=∫S(∇×F)⋅da\oint_{\gamma}\mathbf{F}\cdot d\mathbf{r}=\int_{S}(\nabla\times\mathbf{F})\cdot d\mathbf{a}

which is the correct statement.