Is there a simple way to express a general vector field in terms of the gradient of another (perhaps higher dimensional) function?
Only if it is "exact" (in fact, the definition of "exact" is that it is the derivative of some other function). Even in 2 dimensions, there exist vector fields f(x,y)i+ g(x,y)j that are not graf F for any f.
I know; I am wondering if there is a way to write vector fields that are not gradients of functions in their own dimension as some simple transformation of a gradient of some function of a higher dimension.