Guichard 16.3.6

Will

I'm looking for an $f$ such that $\nabla f = \langle x^2y^3,xy^4 \rangle$. I first integrated $x^2y^3$ with respect to $x$ to get that $f=\frac{x^3y^3}{3}+c(y)$. Next, I integrated $xy^4$ with respect to $y$ to get $\frac{xy^5}{5}+c(x)$. Unfortunately, this leads me to believe that that an $f$ that satisfies $\nabla f = \langle x^2y^3,xy^4 \rangle$ does not exist since there are no $c(x)$ and $c(y)$ that would satisfy this condition.

I've heard that there is a way to check whether or not an $f$ exists before hand. Could someone show this to me so that I can avoid unnecessary integration in the future?

Will

Of course! Given $\nabla f = \langle P, Q \rangle$, an $f$ that satisfies this condition exists if $\frac{\partial}{\partial y}P=\frac{\partial}{\partial x}Q$. In your case, you should have checked to see if $\frac{\partial}{\partial y}x^2y^3$ is equal to $\frac{\partial}{\partial x}xy^4$. You would have seen that $\frac{\partial}{\partial y}x^2y^3=3x^2y^2$ while $\frac{\partial}{\partial x}xy^4=y^4$ and, therefore, $f$ does not exist.