potential function

A scalar function $f$ whose gradient reproduces a given conservative vector field, $\nabla f = \mathbf{F}$ .

Given $\mathbf{F}=(P,Q)$ known to be conservative, a potential function $f$ satisfies $\nabla f = \mathbf{F}$ . It is built by integrating $P$ with respect to $x$ (treating $y$ as constant) to get $f(x,y) = \int P\,dx + g(y)$ , then differentiating with respect to $y$ and matching against $Q$ to solve for $g'(y)$ , and integrating once more. The potential function is never unique — $f$ and $f+C$ for any constant $C$ have the identical gradient — so any one valid answer is "the" potential function up to an additive constant.

Examples

For $\mathbf{F}=(2xy,\ x^2+3y^2)$ , a potential function is $f(x,y) = x^2y + y^3 + C$ .