more generally, a finite measure) that is rotationally invariant, then the convolution of f with satisfies for x outside the support of f In dimension d = 3, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center. When the measure μ is associated to a mass distribution on a sufficiently smooth hypersurface S (a Lyapunov surface of Hölder class C1,α) that divides Rd into two regions D+ and D−, then the Newtonian potential of μ is referred to as a simple layer potential. Simple layer potentials are continuous and solve the Laplace equation except on S. They appear naturally in the study of electrostatics in the context of the electrostatic potential associated to a charge distribution on a closed surface. If is the product of a continuous function on S with the (d − 1)-dimensional Hausdorff measure, then at a point y of S, the normal derivative undergoes a jump discontinuity f(y) when crossing the layer. Furthermore, the normal derivative is of w a well-defined continuous function on S. This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation. See also Double layer potential Green's function Riesz potential Green's function for the three-variable Laplace equation References . . Harmonic functions Isaac Newton Partial differential equations Potential theory Singular integrals