Question

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.) F(x, y, z) = xyz2i + x9yz2j + x9y2zk

Answer 1

We're looking for $f(x,y,z)$ such that $\nabla f(x,y,z)=\vec F(x,y,z)$, which requires

$\dfrac{\partial f}{\partial x}=xyz^2$

$\dfrac{\partial f}{\partial y}=x^9yz^2$

$\dfrac{\partial f}{\partial z}=x^9y^2z$

Integrating both sides of the first PDE wrt $x$ gives

$f(x,y,z)=\dfrac12x^2yz^2+g(y,z)$

Differenting this wrt $y$ gives

$\dfrac{\partial f}{\partial y}=x^9yz^2=\dfrac12x^2z^2+\dfrac{\partial g}{\partial y}$

$\implies\dfrac{\partial g}{\partial y}=\dfrac12x^2z^2(2x^7y-1)$

but we're assuming $g(y,z)$ is a function that doesn't depend on $x$, which is contradicted by this result, and so there is no such $f$ and $\vec F$ is not conservative.