13 Stokes’ Theorem
Theorem 12.1 says that the circulation of F around the boundary of a flat region in \mathbb{R}^2 equals the integral of the curl over the region. Stokes’ theorem extends this to curved surfaces in \mathbb{R}^3: the circulation of F around the boundary of an oriented surface S equals the integral of \operatorname{curl} F over S. The surface can be any smooth oriented surface with boundary — a hemisphere, a piece of a paraboloid, a soap film spanning a wire — and the theorem holds for all of them.
The proof reduces directly to Green’s theorem. Parametrise S by \Phi : D \to \mathbb{R}^3, pull the line integral back to \partial D, apply Green’s theorem on the flat domain D, and track what the integrand becomes under the pullback. The key calculation — that pulling back gives exactly the curl dotted with the surface normal — is the same antisymmetry of the cross product that we computed in Chapter 9.
13.1 The Theorem
A compact oriented surface S \subset \mathbb{R}^3 with boundary inherits an orientation on \partial S from the right-hand rule: if the right thumb points in the direction of \hat n, the fingers curl in the direction of \partial S.
Theorem 13.1 (Stokes’ theorem) Let S \subset \mathbb{R}^3 be a compact oriented C^2 surface with piecewise C^1 boundary \partial S, and F : U \to \mathbb{R}^3 a C^1 vector field on an open set U \supset S. Then \oint_{\partial S} F \cdot d\gamma = \iint_S (\operatorname{curl} F) \cdot d\mathbf{S}.
When S is a flat region \Omega in the xy-plane with \hat n = \hat k, the right side is \iint_\Omega (\partial_x F_2 - \partial_y F_1)\, dA and the left side is \oint_{\partial\Omega} F_1\, dx + F_2\, dy. This is Green’s theorem. Stokes is Green with a curved surface.
Proof. Let \Phi : D \to S be a smooth parametrisation with D
\subset \mathbb{R}^2 a Jordan domain. Writing u = (u_1, u_2) for the parameters, the line integral pulls back as
\oint_{\partial S} F \cdot d\gamma
= \oint_{\partial D} F(\Phi(u)) \cdot D\Phi_u\, \dot{u}\, dt
= \oint_{\partial D} G_1\, du_1 + G_2\, du_2,
where G = (G_1, G_2) = D\Phi^T (F \circ \Phi), i.e.
G_i(u) = \sum_j F_j(\Phi(u))\,\partial_{u_i}\Phi_j(u).
Apply Green’s theorem on D: \oint_{\partial D} G_1\, du_1 + G_2\, du_2 = \iint_D \left(\frac{\partial G_2}{\partial u_1} - \frac{\partial G_1}{\partial u_2}\right) du. Now compute the integrand. By the chain rule, \frac{\partial G_i}{\partial u_\ell} = \sum_j \sum_k \frac{\partial F_j}{\partial x_k} \partial_{u_\ell}\Phi_k\cdot \partial_{u_i}\Phi_j + \sum_j F_j\, \partial_{u_\ell}\partial_{u_i}\Phi_j. The terms involving \partial_{u_\ell}\partial_{u_i}\Phi_j are symmetric in \ell and i, so they cancel in \partial_{u_1}G_2 - \partial_{u_2}G_1. What remains is \partial_{u_1}G_2 - \partial_{u_2}G_1 = \sum_{j,k} \frac{\partial F_j}{\partial x_k} \!\left(\partial_{u_1}\Phi_k\,\partial_{u_2}\Phi_j - \partial_{u_2}\Phi_k\,\partial_{u_1}\Phi_j\right). The antisymmetric factor \partial_{u_1}\Phi_k\,\partial_{u_2}\Phi_j - \partial_{u_2}\Phi_k\,\partial_{u_1}\Phi_j is the (j,k) minor of the 2 \times 3 matrix D\Phi^T. Collecting the terms by which component of \operatorname{curl} F they contribute to: \partial_{u_1}G_2 - \partial_{u_2}G_1 = (\operatorname{curl} F)(\Phi(u)) \cdot (\partial_{u_1}\Phi \times \partial_{u_2}\Phi). Integrating over D gives exactly \iint_S (\operatorname{curl} F) \cdot d\mathbf{S} by the definition of the flux integral. \square
The second-order terms \partial_{u_\ell}\partial_{u_i}\Phi_j cancel by symmetry — this is Theorem 6.2 appearing again, now in the proof of Stokes rather than in the proof of the identities in Chapter 5. The entire calculation is a careful application of the chain rule, with antisymmetry doing all the work.
13.2 Surface Independence
The most striking consequence of Stokes’ theorem is that the flux of \operatorname{curl} F through a surface depends only on its boundary, not on the surface itself.
Theorem 13.2 (Surface independence) Let S_1 and S_2 be two compact oriented surfaces with the same oriented boundary \partial S_1 = \partial S_2 = C. Then for any C^1 vector field F, \iint_{S_1} (\operatorname{curl} F) \cdot d\mathbf{S}_1 = \iint_{S_2} (\operatorname{curl} F) \cdot d\mathbf{S}_2.
Proof. Both sides equal \oint_C F \cdot d\gamma by Stokes. \square
This is the surface analogue of path-independence for conservative fields. Recall from Chapter 9: F = \nabla f implies \int_\gamma F \cdot d\gamma depends only on the endpoints of \gamma. Here: G = \operatorname{curl} F implies \iint_S G \cdot d\mathbf{S} depends only on \partial S.
The analogy with path-independence runs deeper than it might seem. A gradient field is curl-free; path-independence says the integral of a gradient around any closed curve is zero. A curl field is divergence-free; surface-independence says the flux of a curl through any closed surface is zero. Both are consequences of the same structure: consecutive operations in the chain \nabla \to \operatorname{curl} \to \operatorname{div} composed to zero, and the integral theorems translating this into statements about boundaries.
For a closed surface \partial S = \varnothing, Stokes gives \iint_S (\operatorname{curl} F) \cdot d\mathbf{S} = 0, consistent with the Divergence theorem: \iint_{\partial\Omega} (\operatorname{curl} F) \cdot d\mathbf{S} = \iiint_\Omega \operatorname{div}(\operatorname{curl} F)\, dV = 0.
13.3 Conservative Fields via Stokes
Theorem 13.3 (Curl-free on simply connected \Rightarrow conservative) Let U \subset \mathbb{R}^3 be simply connected and F \in C^1(U, \mathbb{R}^3) with \operatorname{curl} F = 0. Then F = \nabla f for some f \in C^1(U).
Proof. By the path-independence theorem (Chapter 9, Theorem Theorem 11.1), it suffices to show \oint_\gamma F \cdot d\gamma = 0 for every closed curve \gamma in U. Since U is simply connected, \gamma bounds a smooth surface S \subset U. Stokes gives \oint_\gamma F \cdot d\gamma = \iint_S (\operatorname{curl} F) \cdot d\mathbf{S} = \iint_S 0\, dS = 0. \qquad\square
Compare with the direct proof in Chapter 9, which constructed the potential explicitly as f(x) = \int_{\gamma_{x_0,x}} F \cdot d\gamma and verified \nabla f = F by computing partial derivatives. That proof works on any connected domain — simple connectivity was not used. The Stokes proof is cleaner, but its simplicity comes at a cost: it uses simple connectivity in a geometric way (every closed curve bounds a surface) rather than building the potential directly. Both proofs are valid; they illuminate different aspects of the same result.
The condition “simply connected” is doing real work. The failure on multiply connected domains is not a defect of the theorem but a reflection of genuine topology. The field F = \left(\frac{-y}{x^2+y^2},\; \frac{x}{x^2+y^2},\; 0\right) on U = \mathbb{R}^3 \setminus \{z\text{-axis}\} satisfies \operatorname{curl} F = 0 everywhere on U, yet \oint_{S^1} F \cdot d\gamma = 2\pi for the unit circle in the xy-plane. The Stokes proof breaks down here because the circle does not bound any surface inside U: any surface it could span would have to cross the missing axis. No such surface exists in U, so the argument \oint_\gamma F \cdot d\gamma = \iint_S \operatorname{curl} F \cdot d\mathbf{S} = 0 has no surface S to use. This is the same obstruction we identified in Chapter 9 from the direct proof — the loop cannot be contracted to a point — now seen through Stokes’ theorem rather than through explicit integration.
13.4 The Chain of Identities
The integral theorems of this part — the fundamental theorem for line integrals, Green’s theorem, Stokes’ theorem, and (next chapter) the Divergence theorem — all have the same structure: an integral over a boundary equals a derivative of the integrand integrated over the interior. The derivatives that appear form a chain: C^\infty(U) \xrightarrow{\;\nabla\;} \{\text{vector fields on }U\} \xrightarrow{\;\operatorname{curl}\;} \{\text{vector fields on }U\} \xrightarrow{\;\operatorname{div}\;} C^\infty(U), in which the composition of any two consecutive maps is zero: \operatorname{curl}(\nabla f) = 0 and \operatorname{div}(\operatorname{curl} F) = 0. These are the identities from Chapter 5, now seen as part of a pattern.
The integral theorems say that integrating over a boundary and applying one step of this chain to the integrand are the same operation. And the topology of the domain determines whether the chain is “exact” — whether every element in the kernel of one map is in the image of the previous one. On \mathbb{R}^3 the answer is yes: every curl-free field is a gradient, and every divergence-free field is a curl. On domains with holes the answer can be no, and the failure is measured by topological invariants of the domain.
This interplay between the calculus of the chain and the topology of the domain is the starting point of algebraic topology and the theory of differential forms — both natural next courses after this one.