EM-16 · Magnetostatics toolkitTool equip

Curl and Stokes' Theorem

The circulation summed around a closed loop equals the curl at every point of the surface it bounds, integrated over that surface. Subdivide the surface to see interior edges cancel and only the boundary remain, and pull out the differential form of Ampère.

Linking the loop and the interior

Subdivide the surface into small cells. On an edge shared by two neighbours, one cell's circulation runs opposite to the other's, so they cancel. What is left over, never cancelled?

Subdivisions N (per side)

Drag to orbit. Buttons set the subdivisions N.

Stokes' theorem

∮F·dl = ∫(∇×F)·dA · loop circulation = surface integral of the curl

Even split into 4 cells, interior edges cancel and only the boundary loop remains.

Neighbors cancel

A bridge between loop and interior

Ampère's law (EM-14) speaks of the circulation ∮B·dl around a closed loop. Stokes' theorem bridges this loop circulation to the curl at every point of the surface the loop bounds, integrated over that surface: ∮F·dl = ∫(∇×F)·dA. The curl ∇×F is a vector measuring how much a field swirls at a point. Where the divergence theorem (EM-07) tied a surface to a volume, Stokes' theorem ties a loop to a surface.

Observe∮F·dl = ∫(∇×F)·dA

The loop circulation equals the surface integral of the curl.

Why it holds — neighbours cancel

Split the surface into small cells and add up each cell's tiny circulation. On an interior edge shared by two cells, one cell goes one way along that edge while its neighbour goes the opposite way. So interior contributions cancel in pairs. The only edges left unpaired are those on the outer boundary. Adding the curl of every cell therefore leaves just the circulation of the boundary loop. One cell's tiny circulation divided by its area is the curl ∇×F.

Choose∇×F = ?

The curl is the circulation per unit area.

Curl, the circulation per unit area

Shrink a cell to nothing and its circulation divided by area becomes the curl ∇×F at a point — the amount of turning there per unit area. The curl is a vector, pointing along the normal of the plane in which the turning is strongest (right hand). It is like a tiny paddlewheel set in water, measuring how much and about which axis it spins at a point. Where divergence measured welling-up, curl measures swirl.

The differential form of Ampère

Put Stokes' theorem into Ampère's law. The loop circulation ∮B·dl is μ₀I, and the interior is ∫(∇×B)·dA. The current I threading the loop is the current density integrated over the surface, ∫J·dA. Then ∫(∇×B)·dA = ∫(μ₀J)·dA holds for any surface, so the integrands must match: ∇×B = μ₀J. The one-line integral form has become a pointwise differential form. It pairs with the divergence theorem giving the differential form of Gauss's law ∇·E=ρ/ε₀ — another of Maxwell's equations.

Fill in∫(∇×B)·dA = ?

By Ampère's law the surface integral becomes the enclosed current.

On your own∇×B = ?

Equal for every surface means the integrands match: ∇×B = μ₀J.

Back to the first screen

On the first screen, the finer you split the surface, the denser the shared interior edges became — yet they all cancelled in pairs, because where one cell turned one way its neighbour turned the other. The only edges left were those on the outer boundary. So the surface integral of every cell's curl exactly equals the circulation of the boundary loop: ∮F·dl = ∫(∇×F)·dA. Lay this bridge onto Ampère's law and out comes the differential form ∇×B = μ₀J.

Stokes' theorem is the identity that the circulation ∮F·dl around a closed loop equals the surface integral of the curl at every point of the surface the loop bounds: ∮F·dl = ∫(∇×F)·dA. It holds because the circulation on edges interior cells share with their neighbours cancels, leaving only the boundary loop. The curl ∇×F is the circulation per unit area — the swirl of a field. Combined with Ampère's law it yields the differential form ∇×B = μ₀J.

Tool equipOnce you equip this tool

Stokes' theorem is the tool for moving freely between loop and surface integrals. Once equipped, the differential form of Ampère's law, ∇×B = μ₀J, is in your hands, and paired with the ∇· of the divergence theorem (EM-07) it completes the three uses of the nabla operator ∇ (∇, ∇·, ∇×). Through Faraday's law (EM-20) and the displacement current (EM-21), it underpins the whole differential form of Maxwell's equations.