Way to express a general vector field

[0rthodontist]

Is there a simple way to express a general vector field in terms of the gradient of another (perhaps higher dimensional) function?

 

[HallsofIvy]

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.

 

[0rthodontist]

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.