Stokes' Theorem — Mathematics Knowledge System

Overview

Stokes' theorem generalizes Green's theorem from a flat region in the plane to a curved surface in 3D space: it relates the circulation of a vector field around the boundary curve ∂S of an oriented surface S to the flux of the field's curl through S itself. It's the 3D statement that 'total spin on the boundary equals total local rotation summed over the surface.'

Interactive Graph

Formal Definition

Definition

Let S be an oriented, piecewise-smooth surface bounded by a simple, closed, piecewise-smooth curve ∂S with orientation induced by S's normal (right-hand rule):

\oint_{\partial S} \mathbf F \cdot d\mathbf r = \iint_S (\nabla \times \mathbf F) \cdot d\mathbf S

Stokes' Theorem

Notation

Notation	Meaning
$\partial S$	The boundary curve of surface S, oriented consistently with S's chosen normal via the right-hand rule

Properties

Surface-independence

\text{If } S_1, S_2 \text{ share the same boundary } \partial S \text{ (with matching orientation), } \iint_{S_1}(\nabla\times\mathbf F)\cdot d\mathbf S = \iint_{S_2}(\nabla\times\mathbf F)\cdot d\mathbf S

Condition: The flux of curl F depends only on the boundary curve, not the specific surface spanning it

Theorems

Theorem 1: Stokes' Theorem

\oint_{\partial S} \mathbf F\cdot d\mathbf r = \iint_S(\nabla\times\mathbf F)\cdot d\mathbf S \text{ for oriented surface } S \text{ with compatibly oriented boundary } \partial S

Corollaries

Follows from Stokes' Theorem

\text{Green's theorem is the special case where } S \text{ is a flat region in the } xy\text{-plane with upward normal, so the surface integral of curl reduces to the 2D scalar curl } \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}

Follows from Stokes' Theorem

\text{If } S \text{ is a closed surface (no boundary, } \partial S = \emptyset\text{), then } \iint_S (\nabla\times\mathbf F)\cdot d\mathbf S = 0

Applications

Faraday's law of induction, ∮_C E·dr = −d/dt∬_S B·dS, is essentially Stokes' theorem applied to relate circulating electric field to changing magnetic flux.

Worked Examples

First compute curl F.
$\nabla \times \mathbf F = (0-0, 0-0, 1-(-1)) = (0,0,2)$
Compute the flux of curl F through S. Since curl F=(0,0,2) is constant, and the hemisphere's flux of a constant vertical field equals that of the flat disk it bounds (same boundary, by surface-independence), use the disk x²+y²≤1, z=0, normal=(0,0,1).
$\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S = \iint_{\text{disk}} (0,0,2)\cdot(0,0,1)\, dA = 2 \cdot \pi(1)^2 = 2\pi$
Now compute the boundary line integral directly: ∂S is r(t)=(cos t, sin t, 0), t∈[0,2π].
$\oint_{\partial S} \mathbf F \cdot d\mathbf r = \int_0^{2\pi} (-\sin t, \cos t, 0)\cdot(-\sin t,\cos t,0)\,dt = \int_0^{2\pi} 1\, dt = 2\pi$

Answer: Both sides equal 2π, verifying Stokes' theorem

Practice Problems

Difficulty 8/10

Use Stokes' theorem to evaluate ∬_S (∇×F)·dS where F=⟨z,x,y⟩ and S is the disk x²+y²≤4 in the plane z=0 (upward normal).

Difficulty 6/10

A vector field F has curl ∇×F = 0 everywhere (irrotational). Using Stokes' theorem, what is the circulation ∮_C F·dr around any closed loop C, and what does this imply physically?

Common Mistakes

Common Mistake

Assuming curl F for F=⟨z,x,y⟩ is zero without computing it carefully.

curl⟨z,x,y⟩ = (∂y/∂y − ∂x/∂z, ∂z/∂z − ∂y/∂x, ∂x/∂x − ∂z/∂y) = (1,1,1), NOT zero — a common error is assuming any 'cyclic-looking' field is curl-free.

Common Mistake

Forgetting that ∂S's orientation must be consistent with S's chosen normal via the right-hand rule.

If S's normal points 'up and out', ∂S must be traversed counterclockwise when viewed from the side the normal points toward — reversing this convention flips the sign of one side of the equation.

Quiz

Stokes' theorem equates the surface integral of curl F over S to:

the volume integral of Fthe line integral of F around the boundary curve ∂Sthe area of Sthe divergence of F

The curl ∇×F of a vector field measures:

how much the field spreads outthe local rotation/circulation of the fieldthe field's magnitudethe field's average value

Historical Background

The theorem is named after Sir George Gabriel Stokes, who set it as an examination question at Cambridge in 1854, though it had been communicated to him earlier by Lord Kelvin (William Thomson) in an 1850 letter — the theorem was known to Kelvin before Stokes popularized it via the exam, a case (like several 'Stigler's law' examples in mathematics) of a result named for someone other than its original discoverer.

1850
Kelvin communicates the result to Stokes in a letter
Lord Kelvin
1854
Stokes poses the theorem as a Smith's Prize examination question at Cambridge, and it becomes known by his name
George Gabriel Stokes

Summary

Stokes' theorem: ∮_{∂S} F·dr = ∬_S (∇×F)·dS, relating boundary circulation to surface flux of curl.
Green's theorem is the flat, xy-plane special case of Stokes' theorem.
The flux of curl F through a surface depends only on the surface's boundary curve, not the specific surface (as long as they share the same boundary and orientation).
Orientation of ∂S must match S's normal via the right-hand rule.

References

WebsiteWikipedia — Stokes' theorem