Differentiable Manifold - Calculus On Manifolds

Calculus On Manifolds

Many of the techniques from multivariate calculus also apply, mutatis mutandis, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the total derivative of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least locally. For example, there are versions of the implicit and inverse function theorems for such functions.

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

• The Lie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
• An affine connection, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.

Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and differential forms. The fundamental theorems of integral calculus in several variables — namely Green's theorem, the divergence theorem, and Stokes' theorem — generalize to a theorem (also called Stokes' theorem) relating the exterior derivative and integration over submanifolds.