3dfd6e…: “Opposite a loud gulf”

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Let $\vec {F}(x,y) = \vector {-6xy^2-3y-3,-6x^2y-3x}$ . Identify whether $\vec {F}$ is a gradient field, and if it is, find a potential function $F$ such that $F(0,0) = 0$ .

You should use the Clairaut gradient test.

$\vec {F}$ is a gradient field. $\vec {F}$ is not a gradient field.

A potential function $F$ such that $F(0,0)=0$ is: $F(x,y) = \answer {-3x^2y^2-3xy-3x}$