conservative vector field

A vector field that is someone's gradient — its line integral between two points never depends on the path taken.

$\mathbf{F}$ is called conservative on a region $D$ when $\mathbf{F} = \nabla f$ for some scalar function $f$ on $D$ , called a potential function for $\mathbf{F}$ . Not every field qualifies. The term "conservative" comes from physics: if $\mathbf{F}$ is a force field with potential energy $-f$ , the work it does moving an object between two points doesn't depend on the path, which is exactly the statement that energy is conserved.

Examples

$\mathbf{F}(x,y)=(2x,2y)$ is conservative because it equals $\nabla f$ for $f(x,y)=x^2+y^2$ .
$\mathbf{F}(x,y)=(-y,x)$ is not conservative — its paddle-wheel test visibly spins everywhere.