The Curl Of Conservative Vector Fields

Recall from the Conservative Vector Fields page that a vector field $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{j}$ on $\mathbb{R}^3$ is said to be conservative if there exists a potential function $\phi$ such that $\mathbf{F} = \nabla \phi$.

We also saw that if $\mathbf{F}$ is a conservative vector field on the domain $D$, then it is necessary that $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, $\frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}$, and $\frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$ for all points $(x, y, z) \in D$.

We will now look at a concrete method to determine if a vector field is conservative provided that the functions $P$, $Q$, and $R$ have continuous partial derivatives.

Definition: If $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ and if $P$, $Q$, and $R$ have continuous partial derivatives on $D$ and $\mathrm{curl} (\mathbf{F}) = \vec{0}$ then $\mathbf{F}$ is a conservative vector field.