  The present paper is a short survey on the mathematical basics of Classical
Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and
spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac
operators, exterior algebra bundles and Connes' differential algebras in the
commutative case, among other elements. We avoid the introduction of principal
bundles and put the emphasis on a module-based approach using Serre-Swan's
theorem, Hermitian structures and module frames. A new proof (due to Harald
Upmeier) of the differential algebra isomorphism between the set of smooth
sections of the exterior algebra bundle and Connes' differential algebra is
presented.
