Section 17.5

Stokes' Theorem

The 3D generalization of Green's Theorem: Relating boundary spin to surface swirl.

Introduction

Imagine a soap film suspended on a wire loop. Stokes' Theorem says that the sum of the "micro-rotations" (curl) across the entire soap film equals the total circulation around the wire loop.

It is the fundamental link between Line Integrals and Surface Integrals.

The Theorem

Statement

Orientation Rule: Use the Right Hand Rule. Fingers curl in direction of implies thumb points in direction of normal for .

Visualizing Boundary & Surface

Interactive: Any Surface Will Do

Worked Examples

Example 1: Using Stokes for a Line Integral

Calculate where and C is intersection of and cylinder .

1. Strategy:

Parameterizing C is annoying. Use Stokes to integrate Curl over the planar surface inside C.

2. Calculate Curl:

3. Surface Integral:

S is part of plane inside cylinder.

Normal vs ?

Upward orientation implies k-component positive. Use .

4. Dot Product:

5. Integrate:

over unit disk.

By symmetry . So integral is Area = .

Example 2: Verify Stokes' Theorem

Verify for over hemisphere .

1. Line Integral (Boundary):

Boundary is unit circle in xy-plane ().

2. Surface Integral (Curl):

Normal to sphere (unit).

By symmetry, x and y integrals die. Result is .

Mass of hemisphere with density z = Center of mass calculation = .