Stokes' Theorem

Stokes' theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the field around the boundary, generalizing Green's theorem to three dimensions.

The Theorem

Theorem11.2Stokes' Theorem

Let $S$ be an oriented piecewise smooth surface in $\mathbb{R}^3$ bounded by a simple, closed, piecewise smooth curve $C = \partial S$ with positive orientation (right-hand rule relative to the surface normal). If $\mathbf{F}$ is a $C^1$ vector field on an open set containing $S$ , then $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$

The theorem states that the circulation of $\mathbf{F}$ around the boundary equals the total "curl flux" through any surface spanning that boundary.

Applications

ExampleVerification of Stokes' theorem

Let $\mathbf{F} = (y, -x, z)$ and $S$ be the upper hemisphere $z = \sqrt{1-x^2-y^2}$ with boundary $C$ the unit circle in the $xy$ -plane.

Line integral: $C: \mathbf{r}(t) = (\cos t, \sin t, 0)$ $\oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} (\sin t)(-\sin t) + (-\cos t)(\cos t)\,dt = \int_0^{2\pi} -1\,dt = -2\pi$

Surface integral: $\nabla \times \mathbf{F} = (0, 0, -2)$ . Using $d\mathbf{S} = (-f_x, -f_y, 1)\,dA$ : $\iint_S (0,0,-2) \cdot d\mathbf{S} = \iint_D -2\,dA = -2\pi \quad \checkmark$

ExampleIndependence of spanning surface

Since $\oint_C \mathbf{F} \cdot d\mathbf{r}$ depends only on $C$ , the surface integral $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ gives the same value for any surface $S$ with $\partial S = C$ . This is often used to simplify computation by choosing a flat surface (like the disk $D$ in the $xy$ -plane) instead of a curved one.

Consequences

Theorem11.3Curl-Free Implies Conservative

On a simply connected domain $D \subseteq \mathbb{R}^3$ , if $\nabla \times \mathbf{F} = \mathbf{0}$ , then $\mathbf{F}$ is conservative. This follows from Stokes' theorem: for any closed curve $C$ bounding a surface $S$ in $D$ , $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \mathbf{0} \cdot d\mathbf{S} = 0$ .

RemarkThe general Stokes' theorem

In the language of differential forms, Stokes' theorem takes the unified form $\int_{\partial M} \omega = \int_M d\omega$ for any differential form $\omega$ on a manifold $M$ with boundary. This single statement encompasses the fundamental theorem of calculus, Green's theorem, the classical Stokes' theorem, and the divergence theorem.