Way to express a general vector field

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A general vector field can be expressed as the gradient of another function only if it is "exact," meaning it is the derivative of some function. In two dimensions, there are vector fields that cannot be represented as the gradient of a function. The discussion explores the possibility of expressing non-gradient vector fields as transformations of gradients from higher-dimensional functions. This raises questions about the relationship between dimensions and the representation of vector fields. Ultimately, the inquiry seeks a method for expressing certain vector fields through higher-dimensional gradients.
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Is there a simple way to express a general vector field in terms of the gradient of another (perhaps higher dimensional) function?
 
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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.
 

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