Decomposition of a Nonlinear Multivariate Function using the Heaviside Step Function

Abstract: Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 in his famous quantum mechanics text has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given in Dirac's text, has been poorly studied. We demonstrate the decomposition of a nonlinear multivariate function into a sum of integrals in which each integrand is composed of a derivative of the function and a direct product of Heaviside step functions. It is an extension of Dirac's single-variable form to that for multiple variables. Moreover, it remains mathematically equivalent to the definition of the Dirac delta function with multiple variables, and offers a mathematically unified expression.