Exercise · ZW-CURL-006 · ★★☆ Intermediate
If \(\mathbf F\) is \(C^1\) and conservative on \(\mathbb R^3\), then \(\operatorname{curl}\mathbf F\) is:
Choose one option.
Show answer
- A. the zero vector at every point
Explanation
If \(\mathbf F=\nabla f\), the mixed partials of \(f\) agree and every difference in the curl cancels: \(\nabla\times\nabla f=\mathbf 0\).
Review the concept: Curl
The curl of a vector field \(\mathbf F=\langle P,Q,R\rangle\) is the vector field that measures, at each point, the field's tendency to rotate, with axis and rate of rotation given by its direction and length.
\[\operatorname{curl}\mathbf F=\nabla\times\mathbf F=\langle R_y-Q_z,\;P_z-R_x,\;Q_x-P_y\rangle\]
More practice problems: curl
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