I have a vector field $$ F = xe^{xy}(xy+1),ye^{xy}(xy+1) $$
and i am trying to show that it is conservative with a potential. $$ \frac{d\Phi}{dx} = xe^{xy}(xy+1) , \frac{d\Phi}{dy} = ye^{xy}(xy+1) $$ $$ \Phi(x,y) = e^{xy}(x^2-\frac{x}{y}+\frac{1}{y^2})+g(y) $$
After this i try to differentiate this with respect of y but i cant really solve it. When i,m trying to use wolfram alpha it says that it cannot be solved with standard mathematical functions. How would i continue from here to know if this vector field is conservative?
It is not a conservative vector field. For a two dimensional vector field, $\vec F = (P, Q) \ $ to be conservative, we must have
$\displaystyle \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$
$\displaystyle \vec F = \big(xe^{xy}(xy+1),ye^{xy}(xy+1)\big)$
$P = xe^{xy}(xy+1), Q = ye^{xy}(xy+1)$
$\displaystyle \frac{\partial Q}{\partial x} = y^2 e^{xy} (xy + 2), \ \frac{\partial P}{\partial y} = x^2 e^{xy} (xy + 2)$
$\displaystyle \frac{\partial Q}{\partial x} \ne \frac{\partial P}{\partial y}$