The most general form of Stokes' Theorem is found in differential geometry:

∫_{∂M} ω = ∫_{M} `d`

`ω`

where `M` is a differential manifold with boundary `∂M`, `ω` is a differential form of order one less than the dimension of M, and `d`

`ω` is its exterior derivative. You can also apply it to the dual of such a form, which is simply a vector field.

The fundamental theorem of calculus, the gradient theorem, the curl theorem, the divergence theorem, Green's Theorem, and the three-dimensional version of Stokes' theorem they teach in undergraduate vector calculus courses are all just special cases of this one big theorem, which says (in what amounts to layman's terms at this point) that the integral of a function around the outside of a thing is equal to the integral of the derivative of the same function throughout the whole thing.

Stokes' Theorem (along with its special cases, which make more obvious sense to our visualizing parts) says a lot about the geometric meaning of the derivative that it is hard to express in words. It allows us to make firm mathematical connections between intuitively obvious ideas about how the universe *should* work and empirical models of how it *actually* works.

See also R. W. R. Darling's Differential Forms and Connections, available from Cambridge University Press, which gives a fine (and mostly coordinate free, which is a blessing) explanation of Stokes' Theorem and the other principles and applications of differential geometry.