Let the equation F(x,y,z)=c represent a set of nonintersecting surfaces. (Each c corresponds to a different surface.) What is the condition that this is a set of equipotential surfaces? In other words, can we define such potential V(c) that its Laplacian vanishes?

solution=
Derivatives of V with respect to space coordinates, can be taken by first taking the derivative of V with respect to c and then the derivative of c with respect to, say, x. Then
grad^2V=V''(c)(gradc)^2+V'(c)grad^2c=0
where grad^2 denotes Laplacian, and prime denotes derivative with respect to c. This leads to the condition
[grad^2c]/[(grad c)^2=-V''(c)/V'(c)=G(c)
i.e. the ratio on the l.h.s. of the equation must be a function of only c. This is the true and the only requirement of the set of surfaces determined by F. The arbitrary function G(c) determines the potential V:
V=A {\int} exp[-{\int}G(c) dc] dc + B
where {\int} denotes the integral sign.. reference ,W. R. Smythe