On Helmholtz's theorem and its interpretations

This paper presents a direct, natural proof of a generalized Helmholtz's theorem for piecewise continuously differentiable vector functions in vector analysis and mathematical physics and its precise statement. Based on the generalized Helmholtz's identity, it is pointed out that Helmholtz's theorem is an operator-based decomposition theorem of a vector function. As a mathematical identity, although it is compatible with some uniqueness theorems (especially those in electromagnetics), it does not indicate directly any uniqueness theorems for boundary value problems. Most existing versions of Helmholtz's theorem are commented. As an important application of the generalized Helmholtz's identity, the definitions of irrotational and solenoidal vector functions are revisited and complete definitions are proposed as a result. The generalized Helmholtz's theorem and the present conclusions should have important indications in vector analysis related disciplines such as electromagnetics.