Stokes’s Theorem and the Divergence Theorem

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 \(S,\) \[ \oiint_S \operatorname{curl}\bm{F} \cdot \bm{N} \,\mathrm{d}S = \oiint_S \operatorname{curl}\bm{F} \cdot \mathrm{d}\bm{S} = \oint_{C} \bm{F}\cdot\mathrm{d}\bm{r} = \oint_{C} \bm{F}\cdot\bm{T} \,\mathrm{d}s \,. \] I.e. the surface integral computing the curl across a surface is equal to the line integral 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 \(E,\) \[ \iiint_E \operatorname{div}\bm{F} \,\mathrm{d}V = \oiint_{S} \bm{F}\cdot\mathrm{d}\bm{S} \,. \] I.e. the integral computing the total divergence within an expanse is equal to the surface integral across its boundary, the flux.