Where can I found the Helmholtz theorem which says that I can always decompose a vector in 2 parts?

There is a theorem of Helmholtz that says a vector field can be written as the sum of the grad of a scalar and curl of a vector. . The proof revolves around constructing a solution then showing it is unique.

 

I suspect, though, that this is at an inappropriate level of complexity.

 

See here:

 

http://mathworld.wolfram.com/HelmholtzsTheorem.html

pick up any mathphys textbook or look at the site that was provide above