Stokes’ Theorem
(or the Kelvin-Stokes Theorem,
or the Curl Theorem) —
for an oriented piecewise-smooth surface \(S\) whose boundary \(C = \partial S\)
is a simple, closed, piecewise-smooth, positively-oriented curve,
and a vector field \(\bm{F}\) whose components
have continuous partial derivatives on a open expanse containing \(S,\)
\[
{\color{silver} \int_a^b\int_c^d \operatorname{curl}\bm{F}\bigl(\bm{r}(s,t)\bigr)\cdot\bigl(\bm{r}_s\times\bm{r}_t\bigr) \,\mathrm{d}s\,\mathrm{d}t
=} \oiint_S \operatorname{curl}\bm{F} \cdot \mathbf{N} \,\mathrm{d}S
%= \oiint_S \operatorname{curl}\bm{F} \cdot \mathrm{d}\bm{S}
= \oint_{C} \bm{F}\cdot\mathrm{d}\bm{r}
{\color{silver} \;\;= \int_a^b \bm{F}\bigl(\bm{r}(t)\bigr)\cdot\bm{r}'(t) \,\mathrm{d}t}
%= \oint_{C} \bm{F}\cdot\bm{T} \,\mathrm{d}s
\,.
\]
I.e. the surface integral that computes the total circulation, the sum of the curl, on a surface
is equal to the line integral that computes the circulation along its boundary.
The Divergence Theorem
(or Gauss’s Theorem,
or Ostrogradsky’s Theorem) —
for a simple expanse \(E\) whose boundary \(S = \partial E\)
is a simple, closed, piecewise-smooth, and positively-oriented (outward) surface,
and for a vector field \(\bm{F}\) whose components
have continuous partial derivatives on a open expanse containing \(E,\)
\[
%{\color{silver} \int\limits_a^b\int\limits_{h_1(x)}^{h_2(x)}\int\limits_{g_1(x,y)}^{g_2(x,y)} \operatorname{div}F(x,y,z) \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x = }
\iiint_E \operatorname{div}\bm{F} \,\mathrm{d}V
= \oiint_{S} \bm{F}\cdot\mathrm{d}\bm{S}
{= \color{silver} \int_a^b\int_c^d \bm{F}\bigl(\bm{r}(s,t)\bigr)\cdot\bigl(\bm{r}_s\times\bm{r}_t\bigr) \,\mathrm{d}s\,\mathrm{d}t}
\,.
\]
I.e. the integral that computes the sum total of the divergence within an expanse
is equal to the surface integral across its boundary, the flux.