Exercise · ZW-CURL-010 · ★★★ Advanced
For \(\mathbf F=\langle xyz,\,x^2,\,y^2\rangle\), \(\operatorname{div}(\operatorname{curl}\mathbf F)\) equals:
Choose one option.
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- C. \(0\)
Explanation
\(\operatorname{curl}\mathbf F=\langle 2y,\,xy,\,2x-xz\rangle\); its divergence is \(0+x+(-x)=0\). Not a coincidence: \(\nabla\cdot(\nabla\times\mathbf F)=0\) for every \(C^2\) field.
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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