Product of Distributions Applied to Discrete Differential Geometry

We propose a formula for evaluating the product of step discontinuous and delta functions. Using tensor calculus and the above proposed formula, we evaluate of the total curvature of a polyhedron vertex where curvature is infinite and total curvature is finite and therefore the Gaussian curvature can be represented by a Dirac delta function. From the above calculation we find the well known deficiency angle formula which gives the discrete curvature of a polyhedron vertex and therefore we find an analytic proof of the known results that the Gauss-Bonnet theorem for smooth surfaces and the Descartes deficiency angle theorem for polyhedron, are the same thing.