Exercise 1918Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Conservative fields in 3D

#conservative#3D#potential

Statement

Show that $\vec F=(yz,xz,xy)$ is conservative and find a potential.

Hint

$\nabla\times\vec F=0$ ; find $\phi$ with $\nabla\phi=\vec F$ .

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Curl. $\nabla\times\vec F=0$ (computed earlier), so $\vec F$ is conservative on $\mathbb{R}^3$ .

Potential. Integrate $yz$ in $x$ : $\phi=xyz+g(y,z)$ . Then $\partial_y\phi=xz+g_y=xz$ , so $g_y=0$ ; similarly $g_z=0$ .

Result. $\phi=xyz$ .

Conclusion. $\vec F=\nabla(xyz)$ .