# Math Help - Determine the Potential Function

1. ## Determine the Potential Function

Folks,

Find the potential function for $F(x,y)=-2xy^3 \vec i -3x^2y^2 \vec j$

IF we take $\phi_x= -2xy^3$ and $\phi_y=-3x^2 y^2$

Integrating the first one wrt x gives

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

Differentiating the RHS wrt y gives

$-3x^2y^2 +k'(y)$ but

$\phi_y=-3x^2 y^2 \implies k'(y)=0 \implies k(y)=C$

$\phi(x,y)=x^2y^3+C$........?

If I apply $\nabla \phi$ is end up with the vector field F above.