Stokes' Theorem and the Divergence Theorem

Green's Theorem had two vector forms in the plane: circulation around a boundary curve equals integrated curl over the region, and flux across the boundary equals integrated divergence. Each lifts into three dimensions. The circulation form becomes Stokes' Theorem, with the flat region replaced by a surface in space and its boundary a space curve. The flux form becomes the Divergence Theorem, with the plane region replaced by a solid and its boundary a closed surface. All four fundamental theorems are the same statement at different dimensions.

Stokes' Theorem

A surface $S$ in space has a boundary curve $C$ , the edge where the surface stops. An orientation of $S$ (a choice of unit normal $\overset{n}{^}$ ) induces a positive orientation of $C$ by the right-hand rule: point the thumb along $\overset{n}{^}$ and the fingers curl in the direction $C$ is traced.

The line integral around the boundary of the tangential component of $F$ equals the surface integral over $S$ of the normal component of $curl F$ . When $S$ is a flat region in the $x y$ -plane with upward normal $\hat{k}$ , the right side is $S \iint (curl F) \cdot \hat{k} d A$ and the statement is exactly Green's Theorem. Stokes' Theorem is Green's Theorem for a curved surface.

Stokes' Theorem: circulation of

F

around the boundary curve

C

equals the flux of

curl F

through the surface

S

; the normal and the boundary orientation obey the right-hand rule.

A consequence: if two surfaces share the same boundary curve $C$ , then the flux of $curl F$ through them is equal, because both equal $C \oint F \cdot d r$ . The flux of a curl depends only on the boundary, not on the surface spanning it. In particular a closed surface has no boundary, so the flux of any curl through a closed surface is zero.

The Divergence Theorem

The flux form of Green's Theorem lifts differently: the plane region becomes a solid $E$ , and its boundary curve becomes the closed boundary surface $S$ , oriented outward.

The outward flux of $F$ across the closed surface equals the triple integral of $div F$ over the solid it bounds. It matches the physical reading of divergence as outflow per unit volume: adding up the local outflow over the whole solid gives the net flow across the surface, because outflow from one interior cell is inflow to the next and cancels, leaving only the boundary.

The Divergence Theorem: outward flux across the closed boundary surface

S

equals the integrated divergence over the enclosed solid

E

; interior outflows cancel, leaving only the boundary.

One theorem in every dimension

The four fundamental theorems of this subject share a form: the integral of a derivative of $F$ over an oriented region equals the integral of $F$ over the oriented boundary of that region.

region \int (derivative of F) = \partial (region) \int F .

Each theorem instantiates it one dimension higher, with the derivative and the boundary changing to match.

The four fundamental theorems nest by dimension: each integrates a derivative of the field over a region and recovers the field on the oriented boundary of that region.

The correspondence is exact once the derivative and boundary are named for each case.

Theorem	Region	Boundary	Derivative of $F$	Statement
FTC	interval $[a, b]$	two endpoints	$F^{'}$	$\int_{a}^{b} F^{'} d x = F (b) - F (a)$
Line integrals	curve $C$	two endpoints	$\nabla f$	$\int_{C} \nabla f \cdot d r = f (B) - f (A)$
Green	plane region $D$	closed curve $\partial D$	$Q_{x} - P_{y}$	$\partial D \oint F \cdot d r = D \iint (Q_{x} - P_{y}) d A$
Stokes	surface $S$	boundary curve $\partial S$	$curl F$	$\partial S \oint F \cdot d r = S \iint curl F \cdot d S$
Divergence	solid $E$	closed surface $\partial E$	$div F$	$\partial E \iint F \cdot d S = E ∭ div F d V$

The three derivative operators — gradient, curl, divergence — are the successive derivatives that appear as the dimension climbs, and the two identities $curl (\nabla f) = 0$ and $div (curl F) = 0$ from the curl-and-divergence lesson say that applying two consecutive operators annihilates the field. That is the statement, in the language of vector calculus, that the boundary of a boundary is empty: the edge of a surface is a closed loop with no endpoints, and the boundary of a solid is a closed surface with no edge.

Differential forms

The single form $\int_{region} d ω = \int_{\partial (region)} ω$ is made precise by the theory of differential forms, where FTC, Green's, Stokes', and the Divergence Theorem are literally one equation, the generalized Stokes' Theorem, applied to forms of degree $0$ , $1$ , and $2$ . Vector calculus is the three-dimensional case of that statement, and the operators gradient, curl, and divergence are the exterior derivative acting on functions, vector fields, and flux fields in turn.¹

Stewart, §16.8 — Stokes' Theorem; §16.9 — The Divergence Theorem; §16.10 — Summary. The right-hand-rule orientation of a surface and its boundary, the surface-independence of the flux of a curl, and the unifying view of the fundamental theorems of vector calculus as one statement relating a region's interior derivative to its oriented boundary. ↩