the 4D Delta function is: As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. or where is a 4-vector field defined in is the 4-divergence of is the component of along direction is a 4D simply connected region of Minkowski spacetime is its 3D boundary with its own 3D volume element is the outward-pointing normal is the 4D differential volume element As a component of the SR Hamilton–Jacobi equation in relativistic analytic mechanics The Hamilton–Jacobi equation (HJE) is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle The generalized relativistic momentum of a particle can be written as where and This is essentially the 4-total momentum of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle , plus momentum due to interaction with the EM 4-vector potential via the particle charge . The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action .