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A scalar potential whose negative spatial gradient is the material acceleration. For a smooth global potential, the spatial curl of the material acceleration vanishes. On a simply connected domain the converse holds under the usual smoothness assumptions. Its physical dimension is squared length divided by squared time.

Equality of line integrals on two selected paths is a necessary test for a conservative vector field, but it is not sufficient. The paths may both miss the component responsible for nonzero curl.

For an explicit example, take $W(x,y,z)=(z-y,x-z,y-x{)}^T$. Its curl is the nonzero constant $2(1,1,1{)}^T$, so it is not a gradient field on ${\mathbb{R}}^3$. Nevertheless, on the plane $x=y$, the field equals $(z-x,x-z,0{)}^T$. Every tangent vector to that plane has equal first and second components, giving zero dot product with $W$. Thus every line integral of $W$ along a path confined to that plane is zero, regardless of endpoints or the path's shape.

In particular, a broken straight path and a genuinely curved path in that plane can have matching line integrals while revealing nothing about global path independence. To establish a conservative vector field, construct a vector-field potential, or verify the appropriate global criterion such as zero curl on a simply connected domain.

The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

The alpha effect is the part of a mean-field electromotive force linear in the mean magnetic field itself, $\mathcal{E}=\mathbf{\alpha }\overline{\mathbf{B}}+\cdots$. The alpha tensor can be anisotropic; it need not be a scalar multiple of the identity. Its curl can couple transverse mean-field components and produce an alpha-squared dynamo.

The vector field $\mathbf{G}=(-y,x,0)/(x^2+y^2)$ has zero curl away from its excluded axis, but has circulation $2\pi$ around a positively oriented circle enclosing that axis. Stokes theorem gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega -S)v_x$. The linearized vortensity amplitude remains constant, because the barotropic pressure and gravitational forces have zero curl and the background vortensity is uniform.

Let a closed oriented surface $S$ lose a patch $H$, leaving $S^{\prime }$. Assume a smooth vector field on the relevant surfaces and that its total curl flux through $S$ is zero. The Stokes theorem gives the title formula when the edge is oriented as the boundary of the remaining surface $S^{\prime }$. The boundary of the removed patch $H$, with its original normal, has the opposite direction and hence the opposite circulation.

For a cylindrical side of radius $R$, if $(\nabla \times \mathbf{v})\cdot \mathbf{n}=-\alpha R$ everywhere on the removed patch, the circulation around the boundary of the remaining surface is $\alpha R\mathrm{area}(H)$. This uses actual curved area, rather than its planar projection, and does not depend on the patch shape. For multiple boundary components, the formula concerns the sum with their induced orientations.

Use circulation to summarize how much a velocity field runs along an oriented closed curve $C$, such as a circle around a fluid vortex. Its definition is the line integralwhere $d\mathbf{l}$ is tangent to $C$ in the chosen direction. Reversing that direction reverses the sign of circulation. Its units are ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, because velocity is multiplied by a length. Counterclockwise circulation in the horizontal plane is positive when the chosen unit normal vector is $+{\mathbf{e}}_z$, according to the right-hand rule.

For solid-body rotation $\mathbf{u}=(-\Omega y,\Omega x,0)$, a counterclockwise circle of radius $R$ has constant tangential velocity $\Omega R$. Multiplying by its circumference givesFor the potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the same circle instead gives ${\Gamma }_C=(K/R)(2\pi R)=2\pi K$, independent of $R$. Stipulate $\Omega =1{\mathrm{s}}^{-1}$, $R=1\mathrm{m}$ and $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$: both velocity fields have ${\Gamma }_C=2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}\approx 6.283185{\mathrm{m}}^2{\mathrm{s}}^{-1}$, although their vorticities in the fluid are $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$ and zero respectively. On a circle of radius $2\mathrm{m}$, their circulations are $8\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$. The comparison plot in vorticity shows this difference in radius dependence.

The bridge between circulation and vorticity is Stokes theorem:Here the velocity field must be continuously differentiable on a neighborhood of the entire spanning surface $S$ inside the fluid, and its boundary orientation must match the unit normal vector $\mathbf{n}$. For solid-body rotation, the disk is admissible and its surface integral is $2\Omega \pi R^2$. For the potential line vortex, the origin is singular and excluded from the fluid: a disk crossing it is not admissible. On an annulus, Stokes theorem instead gives ${\Gamma }_{R_{\mathrm{out}}}-{\Gamma }_{R_{\mathrm{in}}}=0$, with both circulations here measured counterclockwise; the induced inner boundary direction is clockwise. Nonzero circulation therefore does not imply nonzero local curl in the available fluid. ^[ref]

This is useful because Stokes theorem lets a local vorticity measurement determine a loop's circulation when a suitable surface exists, while Kelvin's circulation theorem tracks circulation along a moving material curve. The beauty is the precise link between a local derivative and a global line integral, with the domain deciding which conclusions are valid.

Path independence requires the same line integral for every admissible path between fixed endpoints, not merely a few selected paths. For example, the smooth field $\mathbf{F}=(y-z,0,0)$ vanishes on the entire plane $y=z$. Every path in that plane from $(0,0,0)$ to $(1,1,1)$ therefore has zero integral. Yet the path through $(0,1,0)$ and $(1,1,0)$ has integral one: only the horizontal middle segment contributes. The curl is $(0,-1,-1)$, confirming that the field is not conservative. Special paths can hide a nonconservative component.

In a unit square with prescribed conductive boundary temperatures and impermeable boundaries, the stream function convention $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$ gives linear equations $\Delta \psi =R{\theta }_x$, ${\theta }_t={\psi }_x+\Delta \theta$, with $\psi =\theta =0$ on all sides. The velocity equation follows by taking the vertical-plane curl of Darcy law, and the temperature equation by linearizing the heat equation about $1-z$. Unlike a laterally unbounded Darcy layer, the conducting side boundaries produce a two-dimensional real marginal eigenspace at $R_c=8{\pi }^2$.

For scalar electrical conductivity $\sigma (\mathbf{x})$ and constant permeability ${\mu }_0$, the moving-fluid Ohm law and the quasistatic Ampere-Maxwell law give $\mathbf{E}=-\mathbf{u}\times \mathbf{B}+\eta \nabla \times \mathbf{B}$, where $\eta =1/({\mu }_0\sigma )$ is the magnetic diffusivity. Taking minus the curl and using Faraday's law of induction gives ${\partial }_t\mathbf{B}=\nabla \times (\mathbf{u}\times \mathbf{B})-\nabla \times (\eta \nabla \times \mathbf{B})$. Expanding this expression and imposing $\nabla \cdot \mathbf{B}=0$ yields the title equation. The familiar constant-diffusivity resistive induction equation omits the last term because $\nabla \eta =0$; nonrelativistic motion alone does not establish that simplification.

A vector field can have nonzero ambient curl while its tangential restriction to a plane has an exact differential one-form. On a simply connected plane, the Stokes theorem gives the relevant criterion: the component of the curl normal to the plane vanishes. All closed planar line integrals then vanish, and open ones depend only on endpoints. For example, $(y-z,z-x,x-y)$ restricts on $x-z=1$ to $F\cdot dr=d(x-y)$, despite ambient curl $-2(1,1,1)$.

For an ideal infinitely long solenoid with interior magnetic field $B_0\hat{\mathbf{z}}$ and radius $R$, an axisymmetric Coulomb gauge representative of the magnetic vector potential isThis follows from $2\pi r{A}_{\varphi }=\Phi (r)$ by Stokes theorem. The exterior has zero curl but nonzero circulation around the enclosed magnetic flux. The Coulomb gauge condition alone does not eliminate every residual gauge transformation; symmetry and regularity select this customary representative.

For a continuously differentiable vector field on an oriented surface with compatibly oriented boundary, this vector identity follows by dotting both sides with an arbitrary constant vector $a$ and applying Stokes theorem to $\mathbf{F}\times a$. The curl identity $\nabla \times (a\times \mathbf{F})=a\nabla \cdot \mathbf{F}-(a\cdot \nabla )\mathbf{F}$ gives the surface integrand. In components, $[(d\mathbf{S}\times \nabla )\times \mathbf{F}{]}_i=d{S}_j{\partial }_i{F}_j-d{S}_i{\partial }_j{F}_j$, with the derivatives acting on $\mathbf{F}$. For the position vector field $\mathbf{F}=\mathbf{r}$, it equals $-2d\mathbf{S}$; hence the line integral is minus twice the oriented vector area.

The displayed vector calculus identity follows by contracting the two alternating tensors in the component formula for curl. If both fields have zero divergence, only the two directional derivatives remain. It rewrites the transport terms in the adjoint linearized Navier-Stokes evolution.

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.

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.

Current helicity correlates a magnetic field with its curl, which is proportional to electric current density. It differs from magnetic helicity, which uses a magnetic vector potential. For a solenoidal monochromatic field of wavenumber $k$ in a periodic domain, choose ${\mathbf{a}}_b=\nabla \times \mathbf{b}/k^2$. Then $\nabla \times {\mathbf{a}}_b=\mathbf{b}$ and $H_c=k^2⟨{\mathbf{a}}_b\cdot \mathbf{b}⟩$.

This collects the gradient, divergence, curl and Laplacian formulas in cylindrical and spherical coordinates. Coordinate unit vectors vary with position, so taking their derivatives is essential when translating vector calculus operators from Cartesian coordinates. Curl in spherical coordinates is one of these formulas.

A localized vorticity distribution with nonzero hydrodynamic impulse induces $\mathbf{u}\sim [3\mathbf{n}(\mathbf{I}\cdot \mathbf{n})-\mathbf{I}]/(4\pi r^3)$. This curl-free exterior velocity can remain long-ranged even when the vorticity is localized. In a random ensemble it produces dipolar longitudinal and transverse correlation tails.