Generalized Helmholtz theorem

The widely used Helmholtz theorem separates an arbitrary vector function into two parts: one is an irrotational component which can be expressed by a scalar function and the other is a rotational part which can only be expressed by another vector function. So it is considered as an incomplete projection of a vector function. In this paper, the generalized theorem is proved. In this theorem an arbitrary vector function is separated into three parts; each can be expressed as a scalar function. It is the complete projection for a vector function in vector wave function space.