Curl

For a 3D vector field $\mathbf{F}=⟨P,Q,R⟩$, the curl is

$\nabla \times \mathbf{F}=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\hat{\mathbf{i}}+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\hat{\mathbf{j}}+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\hat{\mathbf{k}}.$

Treating $\nabla =⟨{\partial }_x,{\partial }_y,{\partial }_z⟩$ as a “vector of differential operators,” this is formally a cross product:

$\nabla \times \mathbf{F}=\left|\begin{array}{ccc}\hat{\mathbf{i}}& \hat{\mathbf{j}}& \hat{\mathbf{k}}\\ {\partial }_x& {\partial }_y& {\partial }_z\\ P& Q& R\end{array}\right|.$

For a 2D field $\mathbf{F}=⟨P,Q⟩$ (viewed as 3D with $R=0$, no $z$-dependence): $\nabla \times \mathbf{F}=(Q_x-P_y)\hat{\mathbf{k}}$. The scalar $Q_x-P_y$ is the $\hat{\mathbf{k}}$-component of curl — exactly the quantity appearing in Green’s circulation form.

What curl measures

Curl is a local property — defined pointwise, independent of any loop. At each point, $\nabla \times \mathbf{F}$ is a vector whose direction is the axis around which the field is locally rotating, with magnitude twice the local angular speed.

Concrete pictures:

• Radial field $\mathbf{F}=⟨x,y,0⟩$: $\nabla \times \mathbf{F}=0$. Points outward, no rotation. A paddle wheel placed in this field would translate (be pushed outward) but not spin.
• Rotational field $\mathbf{F}=⟨-y,x,0⟩$: $Q_x-P_y=1-(-1)=2$, so $\nabla \times \mathbf{F}=2\hat{\mathbf{k}}$. Paddle wheel spins counterclockwise around the $z$-axis.
• Constant field: curl is zero.

Curl as circulation density

The precise meaning of “local rotation” comes from Stokes’ theorem applied to a tiny disk of area $\Delta A$ centered at a point with unit normal $\hat{\mathbf{N}}$:

${\oint }_{\partial \text{disk}}\mathbf{F}\cdot d\mathbf{r}\approx (\nabla \times \mathbf{F})\cdot \hat{\mathbf{N}}\Delta A.$

So $(\nabla \times \mathbf{F})\cdot \hat{\mathbf{N}}$ is the circulation per unit area around a tiny loop with normal $\hat{\mathbf{N}}$. Curl is a local circulation density. The direction of curl tells you which axis maximizes that density.

Properties

• Linearity: $\nabla \times (a\mathbf{F}+b\mathbf{G})=a(\nabla \times \mathbf{F})+b(\nabla \times \mathbf{G})$.
• Product rule: $\nabla \times (f\mathbf{F})=f(\nabla \times \mathbf{F})+\nabla f\times \mathbf{F}$.

Curl of a gradient is zero

$\nabla \times (\nabla \varphi )=0$

for any twice-differentiable $\varphi$. Proof: the components are mixed partials of $\varphi$ that cancel by Clairaut’s theorem. This is the cross-partial test for conservativeness.

The converse: on a simply connected region, $\nabla \times \mathbf{F}=0$ implies $\mathbf{F}=\nabla \varphi$ for some $\varphi$. On non-simply-connected regions, curl-free vector fields can still fail to be conservative — see the 2D rotational field on ${\mathbb{R}}^2\setminus \{0\}$.

Connection to Green’s and Stokes’

${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}$

is Stokes’ theorem. Restricted to a flat region in the $xy$-plane, this is Green’s circulation form. Both are statements that closed-loop circulation equals integrated curl over an enclosed surface.

Conservative ⇔ curl-free (on simply connected domains)

This is the slogan. On a simply connected domain:

• $\mathbf{F}$ conservative ⇔ $\nabla \times \mathbf{F}=0$.

The curl is the local obstruction to conservativeness: it tells you, pointwise, whether the field is “trying to circulate.” On a simply connected domain it’s the only obstruction; on others, the topology of the domain adds extra obstructions detected only by closed-loop integrals.

Vector calculus identity: divergence of a curl is zero

$\nabla \cdot (\nabla \times \mathbf{F})=0$

for any $\mathbf{F}$ with continuous second partials. Geometric content: rotation moves things in circles, not in or out of any region — no net source.

Together with $\nabla \times \nabla \varphi =0$, this gives the algebraic skeleton:

• Gradients have zero curl.
• Curls have zero divergence.

On suitable domains the converses hold: zero curl ⇒ gradient field; zero divergence ⇒ curl field (i.e., $\mathbf{F}=\nabla \times \mathbf{G}$ for some vector potential $\mathbf{G}$).

In physics

• Magnetic field: Ampère’s law states $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0{\partial }_t\mathbf{E}$. Magnetic fields curl around currents.
• Faraday’s law: $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$. Changing magnetic flux induces a curling electric field.
• Fluid vorticity: $\mathbf{\omega }=\nabla \times \mathbf{v}$ where $\mathbf{v}$ is fluid velocity. Twice the local angular velocity of fluid elements.

The Maxwell equations and Navier–Stokes all run on divergence and curl.