Stokes' Theorem - General Formulation

General Formulation

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on . First, suppose that α is compactly supported in the domain of a single, oriented coordinate chart {U, φ}. In this case, we define the integral of over as

i.e., via the pullback of α to Rn.

More generally, the integral of over is defined as follows: Let {ψ_i} be a partition of unity associated with a locally finite cover {U_i, φ_i} of (consistently oriented) coordinate charts, then define the integral

where each term in the sum is evaluated by pulling back to Rn as described above. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity.

Stokes' theorem reads: If is an (n − 1)-form with compact support on and denotes the boundary of with its induced orientation, then

Here is the exterior derivative, which is defined using the manifold structure only. On the r.h.s., a circle is sometimes used within the integral sign to stress the fact that the (n-1)-manifold is closed. The r.h.s. of the equation is often used to formulate integral laws; the l.h.s. then leads to equivalent differential formulations (see below).

The theorem is often used in situations where is an embedded oriented submanifold of some bigger manifold on which the form is defined.

A proof becomes particularly simple if the submanifold is a so-called "normal manifold", as in the figure on the r.h.s., which can be segmented into vertical stripes (e.g. parallel to the x_n direction), such that after a partial integration concerning this variable, nontrivial contributions come only from the upper and lower boundary surfaces (coloured in yellow and red, respectively), where the complementary mutual orientations are visible through the arrows.