  The classical Gauss-Green formula for the multidimensional case is generally
stated for $C^{1}$ vector fields and domains with $C^{1}$ boundaries. However,
motivated by the physical solutions with discontinuity/singularity for Partial
Differential Equations (PDEs) and Calculus of Variations, such as nonlinear
hyperbolic conservation laws and Euler-Lagrange equations, the following
fundamental issue arises:
  Does the Gauss-Green formula still hold for vector fields with
discontinuity/singularity (such as divergence-measure fields) and domains with
rough boundaries?
  The objective of this paper is to provide an answer to this issue and to
present a short historical review of the contributions by many mathematicians
spanning more than two centuries, which have made the discovery of the
Gauss-Green formula possible.
