Calculus 1 ended with one theorem: integrating a derivative over an interval only needs the values at the two ends. This course has met three more theorems of the same kind, one for each kind of region: curves, surfaces, solids. Put side by side, they are one idea. (This page connects Stokes' theorem and the divergence theorem with everything before them.)
Key idea
| Theorem | Region | Boundary | Statement |
|---|---|---|---|
| Fundamental theorem of calculus | interval $[a, b]$ | two points | $\int_a^b F'(x)\,dx = F(b) - F(a)$ |
| Fundamental theorem for line integrals | curve $C$ | two endpoints | $\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)$ |
| Green's theorem | plane region $D$ | closed curve | $\oint_C P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA$ |
| Stokes' theorem | surface $S$ | closed curve | $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ |
| Divergence theorem | solid $E$ | closed surface | $\iint_{\partial E} \mathbf{F} \cdot d\mathbf{S} = \iiint_E \nabla \cdot \mathbf{F}\,dV$ |
Why it works
$\nabla \times (\nabla f) = \mathbf{0}$: by Stokes, the flux of $\nabla \times \nabla f$ through any surface is $\oint \nabla f \cdot d\mathbf{r}$ around its edge, which is ${}0$ because the edge is a closed curve (fundamental theorem for line integrals).
$\nabla \cdot (\nabla \times \mathbf{F}) = 0$: by the divergence theorem, its integral over any solid is the flux of $\nabla \times \mathbf{F}$ through the solid's closed surface, which is ${}0$ by Stokes because that surface has no edge. A continuous function whose integral over every small ball is ${}0$ is ${}0$ everywhere.
Orientation is part of every statement: the endpoints come with signs ($+F(b)$, $-F(a)$), the boundary curve of a surface follows the right-hand rule, and a closed surface is oriented outward. Each sign convention is chosen so that the inside edges of small pieces cancel.
Worked examples
Worked example: Choosing the theorem
- A flux through a closed surface is the boundary side of the divergence theorem.
- Replace it by the triple integral of $\nabla \cdot \mathbf{F}$ over the solid cylinder.
Answer: the divergence theorem
Worked example: Checking an identity
- $$\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle = \langle 0 - y, 0 - z, 0 - x \rangle = \langle -y, -z, -x \rangle.$$
- Its divergence: $\frac{\partial}{\partial x}(-y) + \frac{\partial}{\partial y}(-z) + \frac{\partial}{\partial z}(-x) = 0$.
Answer: ${}0$