Stokes' Theorem

Main idea

Stokes' theorem is Green's theorem in 3D: the circulation around a closed curve equals the total spin, the flux of $\operatorname{curl}\vec F$ , through any surface whose edge is that curve. Since every such surface gives the same answer, you're free to pick the easiest one.

{\htmlData{sym=0}{\oint_C\vec F\cdot d\vec r}} = \iint_S\left({\htmlData{sym=1}{\nabla\times\vec F}}\right)\cdot d\vec S

What the symbols mean

$\oint_C\vec F\cdot d\vec r$
Circulation
How much the field pushes you around the boundary curve.
$\nabla\times\vec F$
Curl
The local spin, integrated over the surface.

Explanation

Stokes' theorem simply says that the swirl you feel walking around a loop is the sum of all the little swirls on any surface stretched across that loop. Picture a soap film on a bent wire hoop. On the left, $\oint_C\vec F\cdot d\vec r$ is the circulation: how much the field pushes you along as you walk once around the wire. On the right, $\nabla\times\vec F$ is the spin at each point of the film, and $\iint_S(\ldots)\cdot d\vec S$ adds up how much of that spin points straight through the film. It looks like a lot of notation, but it's the same sentence as Green's theorem: total spin inside equals circulation around the edge.

Take a look at Green's theorem again: $\oint_C P\,dx + Q\,dy = \iint_D(Q_x - P_y)\,dA$ . The spin $Q_x - P_y$ is the $z$ -component of the curl, and a flat region in the $xy$ -plane has normal $\vec k$ . So Green's theorem is just Stokes' theorem for a flat surface. Stokes lets the surface bend and tilt in 3D.

Why it's true. It's the same argument as Green's theorem. Chop the surface into tiny, nearly flat patches. Around each one, the circulation is about (the curl's component through the patch) × (its area). Add them all up, and every interior edge is walked twice in opposite directions and cancels. Only the outer edge $C$ survives.

Orientation must match: walk around $C$ with your head along the normal $\vec n$ , and the surface should be on your left. Equivalently, curl the fingers of your right hand along $C$ ; your thumb points along $\vec n$ .

Any surface works. A circle bounds a flat disk, a hemisphere, or a bag-shaped surface; all give the same flux of curl, because all of them equal the same circulation around the same edge. Pick the easiest, usually the flat one. It makes sense once you see the animation further down: inflate a dome over a fixed circle and the flux of the curl never changes.

A common mistake. It's tempting to compute the flux of $\vec F$ itself through the surface. Stokes' theorem needs the flux of the curl of $\vec F$ . Find $\nabla\times\vec F$ first; often it's a constant vector, and the surface integral becomes (curl · normal) × area.

See it

The field F = ⟨−y, x, z⟩ circulating around the unit circle C, with a dome inflating over it. As you can see, the dome bulges higher and its area grows, yet the flux of curl F = ⟨0, 0, 2⟩ through it stays exactly 2π the whole time. It has to: it always equals the circulation around the same fixed edge. Drag to turn it.

Worked examples

Example 1

Use Stokes' theorem for $\vec F = \langle -y, x, z\rangle$ around the unit circle in the $xy$ -plane (counterclockwise).

1
Find the curl first: $R_y - Q_z = 0$ , $P_z - R_x = 0$ and $Q_x - P_y = 1 - (-1) = 2$ , so $\operatorname{curl}\vec F = \langle 0, 0, 2\rangle$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Evaluate the example: $\vec F = \langle -y, x, z\rangle$ around the unit circle.

Problem 2+5 XP

Find the circulation of $\vec F = \langle z, x, y\rangle$ counterclockwise around the unit circle in the plane $z = 0$ .

Problem 3+5 XP

If $\operatorname{curl}\vec F = \vec 0$ everywhere, what is the circulation around any closed curve?

Problem 4+5 XP

Do two different surfaces with the same boundary curve have the same flux of $\operatorname{curl}\vec F$ ?

Problem 5+5 XP

Find the circulation of $\langle -y, x, 0\rangle$ counterclockwise around the circle of radius 2 in the $xy$ -plane.

Problem 6+5 XP

A long straight wire carries a current $I = 5$ A. Ampère's law, Stokes' theorem applied to the magnetic field, says $\oint_C\vec B\cdot d\vec r = \mu_0 I$ for any loop $C$ around the wire, where $\mu_0 = 4\pi\times10^{-7}$ T·m/A. By symmetry $\vec B$ points along circles around the wire with the same strength $B$ all the way around, so on a circle of radius $r$ the circulation is $B \cdot 2\pi r$ . Find $B$ at $r = 2$ cm, in microtesla.

Problem 7+5 XP

Use Stokes' theorem to find $\oint_C\vec F\cdot d\vec r$ for $\vec F = \langle y, z, x\rangle$ , where $C$ is the triangle with vertices $(1, 0, 0)$ , $(0, 1, 0)$ and $(0, 0, 1)$ , traversed counterclockwise as seen from above.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

The plane $z = y$ slices the cylinder $x^2 + y^2 = 1$ in an ellipse $C$ , oriented counterclockwise as seen from above. Use Stokes' theorem to find $\oint_C\vec F\cdot d\vec r$ for $\vec F = \langle z, 2x, y\rangle$ .

Where it's used

Faraday's law of induction is Stokes' theorem: a changing magnetic flux through a loop drives a circulating electric field around it, which is how generators and transformers work.

The Magnetic Field Around a WireCalc III · Electrical Engineering

Flux Integrals The Divergence Theorem