Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 4733.17.2 Synthetic Differential GeometryThe sense in which we understand the word ‘synthetic’ in this context isthat we place ourselves in the certain category E of manifold–like objectswhere everything is smooth. A main assumption about our category E isthat it is cartesian closed, meaning that functional spaces as well as methodsof classical functional analysis are available.Recall that distributions are usually thought of as very non–smoothfunctions, like the Heaviside function, or the Dirac δ−function. These so–called generalized functions are commonly presented following the Sobolev–Schwartz functional analysis, usually including the integral transforms ofFourier, Laplace, Mellin, Hilbert, Cauchy–Bochner and Poisson. The mainapplication of the theory of generalized functions is the solution of classicalequations of mathematical physics (see e.g., [Vladimirov (1971); Vladimirov(1986)]).On the other hand, there is a viewpoint, firstly stressed by Lawvere[Lawvere (1979)], and fully elaborated by A. Kock [Kock (1981); Kock andReyes (2003)], that distributions are extensive quantities, where functionsare intensive quantities. This viewpoint also makes it quite natural toformulate partial equations of mathematical physics (like classical wave andheat equations) – as ODEs describing the evolution over time of any initialdistributions. For example, the main construction in the theory of the waveequation is the construction of the fundamental solution: the descriptionof the evolution of a point δ−distribution over time.To say that distributions are extensive quantities implies that theytransform covariantly (in a categorical sense). To say that functions areintensive quantities implies that they transform contravariantly. Distributionsare here construed, as linear functionals on the space of (smooth) functions.However, since all functions in the synthetic context are smooth, aswell as continuous, there is no distinction between distributions and Radonmeasures.In the category E one can define the vector space D c(M) ′ of distributionsof compact support on M, for each manifold–like object M ∈ E, namelythe object of −linear maps R M → R. We shall assume that elementarydifferential calculus for functions R → R is available, as in all models ofsynthetic differential geometry (SDG, see [Kock (1981); Moerdijk and Reyes(1991)]). Following Kock [Kock (1981)], we shall also assume some integralcalculus, but only in the weakest possible sense, namely we assume thatfor every ψ : R → R, there is a unique Ψ : R → R with Ψ ′ = ψ and with