Physics often deals with classical models where the dynamical variables are a collection of functions { φ } α over a d-dimensional space/spacetime manifold M where α is the " flavor " index. This involves functionals over the φ' s, functional derivatives, functional integrals, etc. From a functional point of view this is equivalent to working with an infinite-dimensional smooth manifold where its points are an assignment of a function for each α, and the procedure is in analogy with differential geometry where the coordinates for a point x of the manifold M are φ (x).