Etermine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y) = ex cos(y)i + ex sin(y)j

A vector field is conservative *if and only if* we can find a scalar function for which , which means we're looking for a function such that




Integrating both sides of the first PDE with respect to , we get



We're assuming the "constant" of integration is some expression independent of . However, upon differentiating with respect to , we have




which means is *not* a function of alone, contradicting the assumption of the contrary. So our desired does not exist, and therefore is *not* conservative.

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