The Classical Integral Theorems

The first summit of the spine (analysis spine). We state and prove the three theorems of vector calculus — Green's, Stokes', and the divergence (Gauss) theorem — using the operators and integrals of vector-calculus.md and the multiple-integral machinery of multiple-integrals.md. All three share one shape,

$${\int }_{\text{region}}(\text{derivative of }\mathbf{F})={\int }_{\partial (\text{region})}\mathbf{F},$$

which stokes-general.md reveals as a single theorem about differential forms.

References: Spivak, Calculus on Manifolds, ch. 5; Marsden & Tromba; Rudin ch. 10.

1. Orientation Conventions

The theorems are statements about signed quantities, so orientation must be pinned down:

• A region $D\subseteq {\mathbb{R}}^2$ has its boundary curve $\partial D$ oriented counterclockwise (region on the left as you walk).
• A surface $S\subseteq {\mathbb{R}}^3$ with chosen unit normal $\mathbf{n}$ induces a boundary orientation on $\partial S$ by the right-hand rule (fingers curl along $\partial S$, thumb along $\mathbf{n}$).
• A solid $V\subseteq {\mathbb{R}}^3$ has its boundary surface $\partial V$ oriented by the outward normal.

These are the conventions that make all signs in the theorems below consistent; differential-forms.md §5 derives them from a single notion of induced boundary orientation.

2. Green's Theorem

Green's Theorem. Let $D\subseteq {\mathbb{R}}^2$ be a region whose boundary $\partial D$ is a piecewise-$C^1$ simple closed curve, positively (CCW) oriented, and let $P,Q$ be $C^1$ on a neighbourhood of $\overline{D}$. Then $${\oint }_{\partial D}(Pdx+Qdy)={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy.$$

Here $P=P(x,y)$ and $Q=Q(x,y)$ are the components of a vector field $\mathbf{F}=(P,Q)$ — equivalently the coefficients of the $1$-form $Pdx+Qdy$ — so the left side is the circulation ${\oint }_{\partial D}\mathbf{F}\cdot d\mathbf{r}$ and the right side integrates the scalar curl $Q_x-P_y$. They are the integrand, defined at every point of the plane; the figure below shows only the domain $D$ and its oriented boundary, which is why $P,Q$ do not appear in it.

2.1 Proof on a simple region

Take $D$ of "type I", $D=\{(x,y):a\le x\le b,\phi (x)\le y\le \psi (x)\}$. By Fubini (multiple-integrals.md §3) and the one-variable FTC (riemann-integral.md §5),

$${\iint }_D\frac{\partial P}{\partial y}dA={\int }_a^b[P(x,\psi (x))-P(x,\phi (x))]dx=-{\oint }_{\partial D}Pdx,$$

the last equality by matching the top/bottom boundary pieces with their orientations (vertical sides contribute nothing to $\int Pdx$). The symmetric computation on a "type II" region gives ${\iint }_D\frac{\partial Q}{\partial x}dA={\oint }_{\partial D}Qdy$. Adding proves the theorem for regions that are both types. $\square$

2.2 General regions

A region with a more complicated boundary is partitioned into simple pieces by auxiliary cuts. Each cut is traversed twice in opposite directions, so the interior line integrals cancel and only ${\oint }_{\partial D}$ survives, while the area integrals add. This decomposition argument — cut into standard pieces, sum, cancel interior boundaries — recurs in every proof on this page and in stokes-general.md.

2.3 Two readings

Green's theorem is simultaneously the 2-D Stokes ($\oint \mathbf{F}\cdot d\mathbf{r}=\iint (\nabla \times \mathbf{F})\cdot \mathbf{k}dA$, circulation form) and the 2-D divergence theorem ($\oint \mathbf{F}\cdot \mathbf{n}ds=\iint \nabla \cdot \mathbf{F}dA$, flux form), obtained by taking $\mathbf{F}=(P,Q)$ or its $90°$ rotation. Setting $P=-y,Q=x$ gives the area formula $\mathrm{Area}(D)=\frac{1}{2}{\oint }_{\partial D}(xdy-ydx)$.

3. The Divergence (Gauss) Theorem

Divergence Theorem. Let $V\subseteq {\mathbb{R}}^3$ be a compact region whose boundary $\partial V$ is a piecewise-$C^1$ surface with outward unit normal $\mathbf{n}$, and let $\mathbf{F}$ be $C^1$ on a neighbourhood of $V$. Then $${\iiint }_V(\nabla \cdot \mathbf{F})dV={\iint }_{\partial V}\mathbf{F}\cdot \mathbf{n}dS={\iint }_{\partial V}\mathbf{F}\cdot d\mathbf{S}.$$

Proof. By linearity treat $\mathbf{F}=(0,0,R)$. For a region $V=\{(x,y,z):(x,y)\in D,\alpha (x,y)\le z\le \beta (x,y)\}$, Fubini + FTC give

$${\iiint }_V\frac{\partial R}{\partial z}dV={\iint }_D[R(x,y,\beta )-R(x,y,\alpha )]dA,$$

which matches the flux of $(0,0,R)$ through the top and bottom faces (the outward normal supplies the signs; side walls have $\mathbf{n}\perp \hat{z}$, contributing nothing). Summing the three coordinate contributions and decomposing a general $V$ into such pieces (interior faces cancel) completes the proof. $\square$

Meaning. Divergence is source density: the theorem says the total source strength inside $V$ equals the net flux out through its surface — conservation made precise. This is the form of Gauss's law in electromagnetism: $\nabla \cdot \mathbf{E}=\rho /{\epsilon }_0$ integrates to $\oint \mathbf{E}\cdot d\mathbf{S}=Q_{\text{enc}}/{\epsilon }_0$.

3.1 Green's identities

Applying the divergence theorem to $\mathbf{F}=f\nabla g$ and using the product rule (vector-calculus.md §2.2) gives Green's first identity

$${\iiint }_V(f\Delta g+\nabla f\cdot \nabla g)dV={\iint }_{\partial V}f\frac{\partial g}{\partial n}dS,$$

and subtracting the swapped version gives the second identity. These are the multivariable integration-by-parts formulas (riemann-integral.md §6) and are foundational for PDE theory (uniqueness for Laplace/Poisson, Green's functions).

4. Stokes' Theorem (Curl Theorem)

Stokes' Theorem. Let $S\subseteq {\mathbb{R}}^3$ be a piecewise-$C^1$ oriented surface with unit normal $\mathbf{n}$, bounded by a piecewise-$C^1$ simple closed curve $\partial S$ oriented by the right-hand rule, and let $\mathbf{F}$ be $C^1$ near $S$. Then $${\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}={\oint }_{\partial S}\mathbf{F}\cdot d\mathbf{r}.$$

Proof. For a surface given as a graph $z=h(x,y)$ over a region $D$, pull both sides back to $D$ via the parametrization $\mathbf{r}(x,y)=(x,y,h(x,y))$. A direct computation (chain rule for the $z$-dependence) turns the flux of $\nabla \times \mathbf{F}$ through $S$ into a planar line/area integral over $D$ to which Green's theorem (§2) applies, and the planar ${\oint }_{\partial D}$ matches ${\oint }_{\partial S}\mathbf{F}\cdot d\mathbf{r}$ under the parametrization. General surfaces follow by the usual decomposition into graph pieces. $\square$

Meaning. The circulation of $\mathbf{F}$ around the boundary loop equals the total curl (microscopic circulation) threading the surface. With $\mathbf{F}$ the magnetic field this is Ampère's law; it is the integral form of $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Note Stokes' theorem depends only on $\partial S$: any two surfaces with the same boundary give the same flux of $\nabla \times \mathbf{F}$ — a consequence of $\nabla \cdot (\nabla \times \mathbf{F})=0$ plus the divergence theorem.

5. One Pattern, Three Theorems

Lining up the gradient theorem (vector-calculus.md §3.1) with the three above:

$${\int }_{\gamma }\nabla f\cdot d\mathbf{r}=f{|}_{\partial \gamma },{\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}={\oint }_{\partial S}\mathbf{F}\cdot d\mathbf{r},{\iiint }_V(\nabla \cdot \mathbf{F})dV={\iint }_{\partial V}\mathbf{F}\cdot d\mathbf{S}.$$

Read top to bottom these are the cases $\dim =1,2,3$ of

$$\boxed{{\int }_{M}d\omega ={\int }_{\partial M}\omega }$$

for a $0$-, $1$-, $2$-form $\omega$. The operators grad, curl, div are one operator $d$ acting on forms of degree $0,1,2$; the orientation rules of §1 are one rule of induced boundary orientation. Assembling this unification is the job of the final two pages.

6. Where this page is used

• The unification is completed in differential-forms.md (the operator $d$) and stokes-general.md (the theorem).
• The physical specializations (Gauss's law, Ampère's law, Faraday's law) appear in covariant form in SR/fields/electromagnetism.md.

Next: Differential Forms and Exterior Calculus.