Determine if this vector field is a conservative force field?

Question

Does $A(x, \, y) = (3x^2 + 2y^2)i + (4xy + 6y^2)j$ represent a conservative force field? If so, determine the potential $\phi$ in $A = \text{grad} \ \phi. $

From what I understand, we need to find $\phi$ such that $A(x, \, y) = \nabla{\phi} $

$A_x = 3x^2 + 2y^2 = \partial \phi/\partial x \qquad A_y = 4xy + 6y^2 = \partial \phi /\partial y $

If I perform a partial integration of $A_x$ with respect to x

Answer 1

First check that $\frac{\partial}{\partial y}A_x=\frac{\partial}{\partial x}A_y$, i.e. for the mixed second partial derivatives

$$\frac{\partial}{\partial y}A_x=4y;\quad \frac{\partial}{\partial x}A_y=4y$$

By inspection, find that

$$\phi(x,y)=x^3+2y^3+2xy^2+k$$

for the potential function. By itself this shows that $\mathbf{A}(x,y)$ is conservative.