3D conservative-field test

A vector field in space is conservative on a simply connected region exactly when its curl is zero everywhere.

The 3D upgrade of the 2D cross-partials check packages all three cross-partial comparisons into a single vector condition: $\operatorname{curl}\mathbf{F}=\mathbf{0}$ . $\mathbf{F}=\nabla f \Rightarrow \operatorname{curl}\mathbf{F}=\mathbf{0}$ holds unconditionally everywhere, by Clairaut's Theorem. The converse needs the region to be simply connected in the 3D sense — every closed curve inside it must be shrinkable to a point without leaving the region — which is a stronger condition than merely having no missing volume (a solid doughnut has no missing chunk of volume but is not simply connected, since a loop through its hole can't shrink).

Examples

For $\mathbf{F}(x,y,z) = (yz,\, xz,\, xy)$ on $\mathbb{R}^3$ (simply connected): $\operatorname{curl}\mathbf{F}=\mathbf{0}$ , so $\mathbf{F}$ is conservative, with potential function $f(x,y,z) = xyz$ .