Composition with Smooth Maps

Abstract

If f is a map ℝ ⁿ → ℝ^m then a function u in ℝ^m can be pulled back to a function u ∘ f in ℝⁿ, the composition. In Section 6.1 we show that this operation can be defined for all distributions u if f ∈ C^∞ and the differential is surjective. (In Section 8.2 we shall find that the composition can be defined for more general maps f when the location of the singularities of u is known in a rather precise sense.) As an example we discuss in Section 6.2 how powers of real quadratic forms can be used to construct fundamental solutions for homogeneous second order differential operators with real coefficients. In Section 6.3 we use the fact that distributions can be composed with diffeomor-phisms to define distributions on C^∞ manifolds simply as distributions in the local coordinates which behave right when the coordinates are changed. In Section 6.4 we continue the discussion of manifolds by giving a short review of the calculus of differential forms on a manifold, ending up with the Hamilton-Jacobi integration theory for first order differential equations. These results will not be used until Chapter VIII, and the geometrical notions related to the Hamilton-Jacobi theory will be discussed in much greater depth in Chapter XXI.