Conservative Vector Field Tests

Two Dimensions

Condition 1

$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$

Condition 2

$P$ and $Q$ is $C_1$ on an open, simply-connected region.

Conclusion

• If both $(1)$ and $(2)$ are true, then $\mathbf{F}$ is conservative.
• If $(1)$ is true and $(2)$ is false, then the test is inconclusive.
• If $(1)$ is false and $(2)$ is true, then $\mathbf{F}$ is not conservative.

Three Dimensions

All three cross-partials must be true.

$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$ $$\frac{\partial P}{\partial z}=\frac{\partial R}{\partial x}$$ $$\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}$$

i.e. the curl of the vector field must be zero.