Stokes' theorem

Stokes’ theorem is the 3D generalization of Green’s circulation form. It states: the circulation of $\mathbf{F}$ around a closed curve $C$ equals the flux of $\nabla \times \mathbf{F}$ through any oriented surface $S$ whose boundary is $C$.

Statement

Let $S$ be a piecewise-smooth oriented surface in ${\mathbb{R}}^3$ with piecewise-smooth boundary curve $C=\partial S$, oriented compatibly with $S$ by the right-hand rule (thumb along the surface normal $\hat{\mathbf{N}}$, fingers curl in the direction of $C$). Let $\mathbf{F}$ be a $C^1$ vector field on a region containing $S$. Then

${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}.$

A 1D boundary controls a 2D surface in 3D — one dimension is consumed by the curl on the right side. This is the unifying structure of the FTC family.

Special case: planar surface gives Green’s

If $S$ is a flat region in the $xy$-plane with $\hat{\mathbf{N}}=\hat{\mathbf{k}}$, then $(\nabla \times \mathbf{F})\cdot \hat{\mathbf{k}}=Q_x-P_y$, and Stokes’ reduces to Green’s circulation form. Green’s is the planar case of Stokes’.

Why Stokes’ is true (sketch)

Subdivide $S$ into many small patches $P_1,P_2,\dots ,P_N$, each oriented compatibly with $S$. For each patch $P_i$, the local circulation-density interpretation of curl says

${\oint }_{\partial P_i}\mathbf{F}\cdot d\mathbf{r}\approx (\nabla \times \mathbf{F})({\mathbf{x}}_i)\cdot {\hat{\mathbf{N}}}_i\mathrm{Area}(P_i),$

where ${\mathbf{x}}_i$ is a point inside the patch and ${\hat{\mathbf{N}}}_i$ is the patch normal. Sum over all patches:

${\sum }_i{\oint }_{\partial P_i}\mathbf{F}\cdot d\mathbf{r}\approx {\sum }_i(\nabla \times \mathbf{F})({\mathbf{x}}_i)\cdot {\hat{\mathbf{N}}}_i\mathrm{Area}(P_i).$

The right side is a Riemann sum that converges to ${\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}$ as the patches shrink.

The left side: shared edges between adjacent patches cancel pairwise. If patches $P_i$ and $P_j$ share an edge, that edge gets traversed once in $\partial P_i$ and once in $\partial P_j$. Because both patches are oriented compatibly with $S$, the shared edge is traversed in opposite directions in the two boundaries — so the line-integral contributions cancel. After all internal cancellations, the only edges that survive are the ones not shared with any neighboring patch — and those are exactly the edges that make up the outer boundary $C=\partial S$. So

${\sum }_i{\oint }_{\partial P_i}\mathbf{F}\cdot d\mathbf{r}={\oint }_C\mathbf{F}\cdot d\mathbf{r}.$

Combining the two halves: ${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}$ in the limit. The detailed verification of the “circulation ≈ curl·area” approximation on each patch reduces to Green’s theorem applied locally (in a coordinate chart on the surface).

The cancellation idea is the same telescoping that proves the FTC: pair up adjacent contributions, internal terms cancel, only boundary contributions survive.

Surface-independence

Two different oriented surfaces $S_1,S_2$ sharing the same boundary $C$ give the same flux of $\nabla \times \mathbf{F}$:

${\iint }_{S_1}(\nabla \times \mathbf{F})\cdot d\mathbf{S}={\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_{S_2}(\nabla \times \mathbf{F})\cdot d\mathbf{S}.$

Huge in practice: given a complicated surface, swap it for a simpler one with the same boundary. For example, replacing a paraboloid by the flat disk that has the same circular boundary.

Curl-free ⇒ conservative on simply connected regions

If $\nabla \times \mathbf{F}=0$ throughout a simply connected region $\Omega$, then for any closed curve $C\subseteq \Omega$, we can find an oriented surface $S\subseteq \Omega$ with $\partial S=C$, and by Stokes’,

${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_{S}0d\mathbf{S}=0.$

Every closed-loop circulation vanishes, so $\mathbf{F}$ is conservative on $\Omega$.

This finishes the 3D version of the conservative-iff-curl-free criterion. The condition that $\Omega$ is simply connected is exactly what lets us fill in any closed loop with an oriented surface lying in $\Omega$.

Curl as local circulation density

Apply Stokes’ to a tiny disk of area $\Delta A$ centered at a point with normal $\hat{\mathbf{N}}$:

${\oint }_{\partial \text{disk}}\mathbf{F}\cdot d\mathbf{r}\approx (\nabla \times \mathbf{F})\cdot \hat{\mathbf{N}}\Delta A.$

So $(\nabla \times \mathbf{F})\cdot \hat{\mathbf{N}}$ is the circulation per unit area around an infinitesimal loop with that normal direction. The vector $\nabla \times \mathbf{F}$ at each point has direction maximizing this density.

This is the precise statement of “curl = local rotation” that’s invoked when introducing curl heuristically.

Worked example: hemisphere

$\mathbf{F}=⟨y,-x,0⟩$, $S=$ upper unit hemisphere $x^2+y^2+z^2=1$, $z\ge 0$, $C=$ unit circle in $xy$-plane counterclockwise viewed from above.

Left side: parameterize $C$ by $\mathbf{r}(t)=⟨\cos t,\sin t,0⟩$, ${\mathbf{r}}^{\prime }(t)=⟨-\sin t,\cos t,0⟩$, $\mathbf{F}(\mathbf{r}(t))=⟨\sin t,-\cos t,0⟩$. Dot: $-{\sin }^{2}t-{\cos }^{2}t=-1$. ${\int }_0^{2\pi }(-1)dt=-2\pi$.

Right side: $\nabla \times \mathbf{F}=\nabla \times ⟨y,-x,0⟩=⟨0,0,-1-1⟩=⟨0,0,-2⟩$. By surface-independence, swap the hemisphere for the flat disk $D$ in the $xy$-plane: $\hat{\mathbf{N}}=\hat{\mathbf{k}}$, so $(\nabla \times \mathbf{F})\cdot \hat{\mathbf{N}}=-2$. Flux $=-2\cdot \mathrm{Area}(D)=-2\pi$. ✓

Trading complicated surfaces for simple ones

$\mathbf{F}=⟨-y^3,x^3,-z^3⟩$, $S=$ paraboloid $z=4-x^2-y^2$ with $z\ge 0$, oriented upward.

Boundary $C$: the circle $x^2+y^2=4$ in $z=0$, counterclockwise.

By Stokes’, ${\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}={\oint }_C\mathbf{F}\cdot d\mathbf{r}$. Parameterize $C$ by $\mathbf{r}(t)=⟨2\cos t,2\sin t,0⟩$. Compute as a line integral; result $24\pi$.

Direct computation of the paraboloid flux would have been substantially nastier. Use Stokes’ to deform the surface to its simplest equivalent.

Connection to FTC family

Theorem	Boundary	Interior
FTC	$F(b)-F(a)$	$\int F^{\prime }dx$
FTLI	$\varphi (B)-\varphi (A)$	${\int }_C\nabla \varphi \cdot d\mathbf{r}$
Green’s (circ)	$\oint \mathbf{F}\cdot d\mathbf{r}$	$\iint (Q_x-P_y)dA$
Stokes’	$\oint \mathbf{F}\cdot d\mathbf{r}$	$\iint (\nabla \times \mathbf{F})\cdot d\mathbf{S}$
Divergence	$\iint \mathbf{F}\cdot d\mathbf{S}$	$\iiint \nabla \cdot \mathbf{F}dV$

Stokes’ is one rung on the ladder. Each says “integral of derivative on interior = boundary integral on boundary.”