Ivancevic_Applied-Diff-Geom

476 Applied Differential Geometry: A Modern Introductionwhere ψ ′ denotes the ordinary derivative of the function ψ.The following Proposition is an application of the covariant functoralityof the functor D ′ c, which will be used in connection with the wave (andheat) equations in dimension 2. We consider the (orthogonal) projectionp : R 3 → R 2 onto the xy−plane; ∆ denotes the Laplace operator in therelevant R n , so for R 3 , ∆ is ∂ 2 /∂x 2 +∂ 2 /∂y 2 +∂ 2 /∂z 2 . For any distributionS (of compact support) on R 3 ,p ∗ (∆(S)) = ∆(p ∗ (S)).This follows from the fact that for any ψ : R 2 → R,namely, ∂ 2 ψ/∂x 2 + ∂ 2 ψ/∂y 2 .∆(p ∗ ψ)) = p ∗ (∆(ψ)),3.17.2.2 Synthetic Calculus in Euclidean SpacesRecall that a vector space E (which in the present context means anR−module) is called Euclidean if differential and integral calculus for functionsR → E is available (see [Kock (1981); Moerdijk and Reyes (1991)]).The coordinate vector spaces are Euclidean, but so are also the vector spacesR M , and D c(M) ′ for any M. To describe for instance the ‘time–derivative’f˙of a function f : R → D c(M), ′ we put< ˙ f(t), ψ >= d dt < f(t), ψ > .Similarly, from the integration Axiom for R, one immediately proves thatD c(M) ′ satisfies the integration Axiom, in the sense that for any h : R →D c(M), ′ there exists a unique H : R → D c(M) ′ satisfying H(0) = 0 andH∫ ′ (t) = h(t) for all t. In particular, if h : R → D c(M), ′ the ‘integral’bh(u) du = H(b) − H(a) makes sense, and the Fundamental Theorem ofaCalculus holds.As a particular case of special importance, we consider a linear vector–field on a Euclidean R−module V . To say that the vector field is linear isto say that its principal–part formation V → V is a linear map, Γ, say. Wehave then the following version of a classical result. By a formal solutionfor an ordinary differential equation, we mean a solution defined on the setD ∞ of nilpotent elements in R (these form a subgroup of (R, +)).Let a linear vector–field on a Euclidean vector space V be given by thelinear map Γ : V → V . Then the unique formal solution of the correspond-