EM-16 · Magnetostatics toolkitTool equip

Curl and Stokes' Theorem

The circulation added up around a closed loop equals the curl at every point of a surface bounded by that loop, integrated over the surface. Splitting the surface into cells, you see that neighboring edges cancel and only the outer boundary remains, and from this you draw the differential form of Ampère's law.

Link the loop and what lies inside it

The gold loop is the edge of the surface, and the cyan grid cuts the surface into small cells. Each cell has a small rotation marker; as you increase the divisions with the buttons, on an edge shared by two neighboring cells the circulation of one cell and that of the next run in opposite directions, like the pair of gray arrows, and cancel. What remains uncanceled to the end is only the circulation around the gold boundary loop.

Subdivisions N (per side)

Drag to rotate; the buttons change the number of divisions per edge (N).

Stokes' theorem

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

Even split into 4 cells, the inner edges cancel in pairs and only the boundary loop remains.

Neighbors cancel

A bridge between loop and interior

Ampère's law (EM-14) speaks of the circulation around a closed loop, ∮B·dl. Stokes' theorem links the two by saying that this circulation equals the curl at every point of a surface bounded by the loop, integrated over that surface. ∮F·dl = ∫(∇×F)·dA. The curl ∇×F is a vector that measures how much a field swirls at a point. For any surface spanning the same loop, flat or bulging, the result is the same, and this property leads later to the reason displacement current is needed. If the divergence theorem (EM-07) linked a surface and a volume, Stokes' theorem links a loop and a surface.

Observe∮F·dl = ∫(∇×F)·dA

The loop circulation equals the surface integral of the curl inside.

Why it holds — neighbours cancel

Split the surface into small cells, find each cell's small circulation, and add them all. On an inner edge shared by two cells, if one cell goes around that edge one way, the neighboring cell goes around the same edge the other way. So the contributions of the inner edges cancel in pairs, and the only edges left without a partner are those on the outer boundary. In the end, adding the circulation of every cell leaves only the circulation around the boundary loop. A cell's small circulation divided by its area is exactly the curl ∇×F.

Choose∇×F = ?

Curl is circulation per unit area.

Curl, the circulation per unit area

Shrink a cell without limit, and its circulation divided by its area becomes the curl ∇×F at a point, the amount of turning at that point per unit area. Curl is a vector, pointing along the normal of the plane in which the field turns most strongly, with its direction set by the right hand. It is like measuring how much, and about which axis, a tiny paddle wheel floating in water spins at a point. If divergence measured springing out, curl measures swirling.

The differential form of Ampère

Put Stokes' theorem into Ampère's law. The loop circulation ∮B·dl is μ₀I, and the inside is ∫(∇×B)·dA. The current I through the loop is the current density integrated over the surface, ∫J·dA. Then ∫(∇×B)·dA = ∫(μ₀J)·dA holds for any surface you choose, so the two integrands must be equal. ∇×B = μ₀J. The one-line integral form has turned into a differential form that holds at every point. It is another of Maxwell's equations, the partner of the differential form of Gauss's law, ∇·E=ρ/ε₀, that the divergence theorem gave.

Fill in∫(∇×B)·dA = ?

Ampère's law turns the surface integral into the enclosed current.

On your own∇×B = ?

The integrals agree over every surface, so the integrands agree. ∇×B = μ₀J.

Back to the first screen

On the first screen, the finer you split the surface, the more shared edges crowded the inside, but they all canceled in pairs, because when one cell turned one way its neighbor turned the other. Only the edges of the outer boundary survived to the end, and that is why the surface integral that adds up every cell's curl exactly equals the circulation around the boundary loop. ∮F·dl = ∫(∇×F)·dA. Set this bridge on top of Ampère's law and the differential form ∇×B = μ₀J comes out.

Stokes' theorem is the identity that the circulation ∮F·dl around a closed loop equals the curl integrated over the surface at every point of a surface bounded by that loop. ∮F·dl = ∫(∇×F)·dA. It holds because the circulations on edges the inner cells share with their neighbors cancel and only the boundary loop remains. The curl ∇×F is circulation per unit area, the swirl of the field. Combined with Ampère's law, it gives the differential form ∇×B = μ₀J.

Tool equipOnce you equip this tool

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