Conservative Vector Fields

A conservative vector field describes a force or field in which the work done between two points depends only on the initial and final positions. For a ball moving in Earth’s gravitational field, gravity performs work determined by the difference in height, regardless of whether the ball moves vertically or follows a curved trajectory.

A vector field is conservative if it can be expressed as the gradient of a scalar potential function, f. In two dimensions, this is written as

\begin{equation*}\mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle=\nabla f\end{equation*}

where

\begin{equation*}P=\jfrac{\partial f}{\partial x}, \qquad Q=\jfrac{\partial f}{\partial y}\end{equation*}

Thus, the components of the vector field are derived from the same potential function. Because P and Q come from the same function, their mixed partial derivatives must be equal. This condition follows from Clairaut’s theorem, which states that mixed partial derivatives are equal when the second partial derivatives are continuous. So, equality of these derivatives is a necessary condition for a field to be conservative.

For the test to be sufficient, the domain of the vector field must be open and simply connected. An open domain contains neighborhoods around each of its points, while a simply connected domain has no holes or gaps. If these conditions are satisfied and the mixed partial derivatives are equal, the vector field is conservative.

In such fields, line integrals can be evaluated by comparing potential values at endpoints:

\begin{equation*}\int_C \mathbf{F}\cdot d\mathbf{r}=f(B)-f(A)\end{equation*}

where A and B are the initial and final points.