Memes

Stokes' theorem. Almost the same thing as the high school one. It generalizes the fundamental theorem of calculus to arbitrary smooth manifolds. In the case that M is the interval [a, x] and ω is the differential 1-form f(t)dt on M, one has dω = f'(t)dt and ∂M is the oriented tuple {+x, -a}. Integrating f(t)dt over a finite set of oriented points is the same as evaluating at each point and summing, with negatively-oriented points getting a negative sign. Then Stokes' theorem as written says that f(x) - f(a) = integral from a to x of f'(t) dt.

It's the most general form of Stokes' theorem that the integral of a differential form over the boundary of an volume and the integral of an exterior derivative of this form over that volume are the same. It covers a lot of classic formulas from the fundamental theorem of calculus to Green's theorem, Gauss' theorem and classic Stokes' theorem.

Same as high school but fancier?