## Problem on finding the potential function of a vector field

$\mathbf{F}(x,y) = 2 \mathbf{i} + 3 \mathbf{j}$ is a conservative vector field. Find a potential function for it.
• ## Solution

Recall that
To find the potential function $\phi(x,y)$, we write out \begin{align} \mathbf{F}(x,y) &= \nabla \phi(x,y) \\ 2 \mathbf{i} + 3 \mathbf{j} &= \partial_x \phi(x,y) \ \mathbf{i} + \partial_y \phi(x,y) \ \mathbf{j} \end{align}
Thus, $\partial_x \phi(x,y) = 2$ and $\partial_y \phi(x,y) = 3$.
Integrating, we get $\phi(x,y) = 2 x + g(y)$ and $\phi(x,y) = 3 y + h(x)$.
Combining, a possible potential function for $\mathbf{F}(x,y)$ is $\phi(x,y) =2x + 3y$.