What does Stokes' theorem state and how is it used to derive Maxwell's curl equations?

In one line

Stokes' theorem states that the line integral of a vector field around a closed loop equals the surface integral of its curl over any surface bounded by that loop.

Explained simply

Tile a soap film stretched on a wire ring with many tiny loops. The swirl of each tiny loop is the curl. Where two loops share an edge, they travel it in opposite directions, so these cancel. Only the outer edge, the wire ring, is left. So the total swirl inside equals the circulation around the rim.

How it works

1. Divide the open surface \(S\) into tiny patches \(\Delta S\). Circulation around each patch \(\approx(\nabla\times\vec{A})\cdot\Delta\vec{S}\).
2. Add over all patches. Internal edges cancel in pairs because they are traversed in opposite directions.
3. Only the boundary curve \(C\) remains: \(\oint_C\vec{A}\cdot d\vec{l} = \int_S(\nabla\times\vec{A})\cdot d\vec{S}\).
4. Direction rule: curl your right-hand fingers along \(C\); your thumb gives the direction of \(d\vec{S}\).
5. Use: applied to Faraday's law it gives \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\); applied to Ampere's law it gives \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\).

Key formula

\(\oint_C\vec{A}\cdot d\vec{l} = \int_S(\nabla\times\vec{A})\cdot d\vec{S}\). \(C\): closed curve, \(d\vec{l}\): line element along \(C\) (m), \(S\): open surface bounded by \(C\), \(d\vec{S}\): area element (m²).

Quick example

Verify for \(\vec{A} = -y\hat{i} + x\hat{j}\) on a circle of radius \(a\) in the xy-plane. Line integral: with \(x = a\cos t, y = a\sin t\), \(\vec{A}\cdot d\vec{l} = a^2(\sin^2t + \cos^2t)\,dt = a^2\,dt\), so the total is \(2\pi a^2\). Surface integral: \(\nabla\times\vec{A} = 2\hat{k}\), over area \(\pi a^2\) gives \(2\pi a^2\). Both agree.

Common mistakes

• Using a closed surface; Stokes' theorem needs an open surface bounded by the curve.
• Choosing the normal direction inconsistent with the loop direction.
• Confusing it with the divergence theorem, which relates surface and volume integrals.

How MU asks it

Asked as "State Stokes' theorem" (3 marks) or used in "Derive Maxwell's third and fourth equations" (5 marks). Write the statement in words and symbols, sketch a loop with a surface and normal, and show where it converts integral to point form.