Curl ($\nabla \times$)

The curl of $F$ has components

For a continuously differentiable vector field in three-dimensional Euclidean space, the component of its curl along a unit normal vector is the limiting circulation per unit area around a small loop in the perpendicular plane. The loop direction and the normal obey the right-hand rule. Stokes theorem gives the corresponding relation for a finite surface.

For the rigidly rotating velocity field $G=(-\Omega y,\Omega x,0)$, direct differentiation gives $\nabla \times G=(0,0,2\Omega )$. A circle of radius $R$ has tangential speed $\Omega R$ and circumference $2\pi R$, so its circulation is $2\pi \Omega R^2$, twice the angular velocity times the enclosed area. The factor of two distinguishes vorticity from rigid-body angular velocity.

Curved integral curves alone do not demonstrate nonzero curl. The field $H=(x,-y,0)$ has zero curl, yet its trajectories satisfy $x=x_0{e}^t$, $y=y_0{e}^{-t}$ and hence $xy=x_0{y}_0$: away from the axes they are hyperbolas. The curl measures local circulation of the field, not the curvature of an individual trajectory.

The divergence theorem applied to $G\times c$ for every constant vector $c$ gives ${\int }_V\nabla \times GdV={\int }_{\partial V}n\times GdS$, with outward unit normal $n$. Indeed $\nabla \cdot (G\times c)=c\cdot (\nabla \times G)$, while $(G\times c)\cdot n=c\cdot (n\times G)$. Equality of the scalar projections for every $c$ proves the vector identity. Reversing the surface cross product reverses the sign.

For a continuously differentiable scalar field $\varphi$ and vector field $\mathbf{F}$, the curl satisfiesIndeed, with fixed Cartesian basis vectors ${\mathbf{e}}_i$, the Cartesian expression for the curl and the product rule giveThe first sum is the cross product of the gradient of $\varphi$ with $\mathbf{F}$; the second is $\varphi$ times the curl of $\mathbf{F}$. The formula is useful when applying the Stokes theorem to a weighted vector field.

For a twice continuously differentiable function, the curl of its gradient vanishes. In components, ${ϵ}_{ijk}{\partial }_j{\partial }_{k}f=0$ because the second derivatives are symmetric in $j,k$ while the Levi-Civita symbol is antisymmetric. Thus adding a gradient to a vector potential preserves the represented curl. No topological assumption is needed for this identity, unlike the converse assertion that every curl-free vector field has a global potential.

A continuously differentiable vector field on an open subset of three-dimensional Euclidean space is irrotational when its curl is zero. Under the Euclidean identification $\alpha =X_{i}d{x}^i$, this says that the differential one-form $\alpha$ is a closed differential form. The Poincaré lemma gives a potential of a conservative vector field locally. A global potential additionally requires zero line integrals around every closed curve; simply connected domains ensures this. On the punctured plane extended in the third direction, $X=(-y,x,0)/(x^2+y^2)$ has zero curl but integral $2\pi$ around a circle, so it is locally conservative and has no single-valued global potential.

A vector potential for a divergence-free vector field $F$ is a vector field $A$ satisfying $F=\nabla \times A$. It is not unique because adding a gradient leaves its curl unchanged.

Let $D$ be bounded, connected and simply connected, and let $u$ be a smooth divergence-free field. Suppose a sufficiently smooth vector field $A$ satisfies the displayed Poisson and boundary conditions. Divergence gives $\Delta \mathrm{div}A=0$ with zero boundary value, so harmonic uniqueness gives $\mathrm{div}A=0$. The vector Laplacian identity then implies $\nabla \times (\nabla \times A-u)=0$.

On a simply connected domain, define a scalar potential by integrating this curl-free difference along a path from a fixed point. Vanishing circulation around contractible loops proves path independence. Thus the difference is $\nabla \chi$; its divergence and boundary normal component give $\Delta \chi =0$ and ${\partial }_n\chi =0$. Green's identity yields ${\int }_D|\nabla \chi {|}^2=0$, proving the reconstruction. Additional conditions on $A$ can specify its own boundary problem; the argument assumes its existence and proves that its curl is exactly $u$. Simple connectivity is essential for eliminating possible non-gradient harmonic circulation fields.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

For a twice differentiable vector field $A$,