Guichard 16.3.6

Will
Sun, 04 Feb 2018 23:11:59 +0000
Ask and Answer

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?