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.