Stokes' Theorem - Introduction

Introduction

The fundamental theorem of calculus states that the integral of a function f over the interval can be calculated by finding an antiderivative F of f:

Stokes' theorem is a vast generalization of this theorem in the following sense.

• By the choice of F, . In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form, i.e. function, F: in other words, that dF = f dx. The general Stokes theorem applies to higher differential forms instead of F.
• A closed interval is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and the form has to be compactly supported in order to give a well-defined integral.
• The two points a and b form the boundary of the open interval. More generally, Stokes' theorem applies to oriented manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientation from that of the manifold. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, a inherits the opposite orientation as b, as they are at opposite ends of the interval. So, "integrating" F over two boundary points a, b is taking the difference F(b) − F(a).

In even simpler terms, one can consider that points can be thought of as the boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral (f dx = dF) over a 1-dimensional manifolds by considering the anti-derivative (F) at the 0-dimensional boundaries, one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (dω) over n-dimensional manifolds (Ω) by considering the anti-derivative (ω) at the (n-1)-dimensional boundaries (dΩ) of the manifold.

So the fundamental theorem reads: