Stokes' Theorem
Stokes' theorem relates the line integral of a vector field around a closed curve to the surface integral of its curl over the enclosed surface.
The Formula
Stokes' theorem is the 3D generalization of Green's theorem. It relates the line integral of a vector field F around a closed curve C to the surface integral of the curl of F over any surface S whose boundary is C. This is one of the fundamental theorems of vector calculus, alongside Green's theorem and the Divergence theorem.
Variables
| Symbol | Meaning |
|---|---|
| C | Positively oriented closed curve (boundary of S) |
| S | Oriented surface with boundary C |
| F | Vector field F = (P, Q, R) with continuous partial derivatives |
| dr | Infinitesimal tangent vector along C |
| ∇ × F | Curl of F = (R_y − Q_z, P_z − R_x, Q_x − P_y) |
| dS | Oriented area element of surface S |
The key insight: you can replace any surface bounded by C with any other surface bounded by C — the integral will be the same. If the curl of F is zero everywhere (∇ × F = 0), the vector field is conservative and all path integrals depend only on endpoints.
Example 1 — Hemisphere Calculation
Evaluate ∮C F · dr where F = (−y, x, z²) and C is the circle x² + y² = 1 in the xy-plane (z = 0), traversed counterclockwise.
Compute curl F: ∇ × F = (0−0, 0−0, 1−(−1)) = (0, 0, 2)
Take S = disk in xy-plane: dS = (0, 0, 1) dA, so (∇ × F) · dS = 2 dA
&iint;S 2 dA = 2 × area of unit disk = 2 × π(1)²
∮C F · dr = 2π
Example 2 — Conservative Field Check
Is F = (2xy, x² + z, y) conservative? Use the curl test.
∇ × F = (∂y/∂y − ∂(x²+z)/∂z, ∂(2xy)/∂z − ∂y/∂x, ∂(x²+z)/∂x − ∂(2xy)/∂y)
= (1 − 1, 0 − 0, 2x − 2x)
curl F = (0, 0, 0) — F is conservative! The line integral around any closed path is zero.
When to Use It
Use Stokes' theorem when:
- The line integral around a closed curve is difficult but the curl surface integral is easier
- Proving that a vector field is conservative (curl = 0)
- Deriving Faraday's law of electromagnetic induction in integral form from the differential form
- Ampere's law in electromagnetism: ∮ B · dl = μ&sub0; × enclosed current
- Studying fluid circulation and vorticity in fluid mechanics