Conservative Fields & Potential Functions

Test for a potential, recover f, integrate by endpoints.

The idea

A vector field $\mathbf{F}$ is conservative if it is the gradient of some scalar function: $\mathbf{F} = \nabla f$. Such an $f$ is a potential for $\mathbf{F}$. For conservative fields, line integrals reduce to an endpoint calculation.

Theorem (Fundamental Theorem for Line Integrals).

If $\mathbf{F} = \nabla f$ with $\nabla f$ continuous, and $C$ is a smooth curve from a point $A$ to a point $B$, then $\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A).$

This is the Fundamental Theorem of Calculus with the interval replaced by a curve: integrating a derivative along a path returns the net change of the function, and the route taken drops out. A general field's line integral depends on the whole path; a conservative field's depends only on where the path starts and ends.

Two computations make the theorem usable. To decide whether $\mathbf{F} = \langle P, Q\rangle$ has a potential, compare $\partial P/\partial y$ with $\partial Q/\partial x$: for a genuine potential both equal the mixed partial $f_{xy}$, so they must agree, and on a simply connected domain their agreement guarantees a potential. To find the potential, undo the two partial derivatives one at a time — integrate $P$ in $x$, then adjust by a function of $y$ so that $f_{y} = Q$.

Ways to work on it

• Walkthrough. The mixed-partials test, recovering a potential, and the Fundamental Theorem for Line Integrals.
• Practice. Decide whether a field is conservative.
• Hardest. Recover a potential and evaluate a line integral by its endpoints.

Not sure where to start? Take the ten-question placement test.