Exercise · ZW-DIVERGENCE-010 · ★★★ Advanced
For \(\mathbf F=(x^2+y^2+z^2)\langle x,\,y,\,z\rangle\), \(\operatorname{div}\mathbf F\) is:
Choose one option.
Show answer
- D. \(5(x^2+y^2+z^2)\)
Explanation
Each component gives, e.g., \(\partial_x\big((x^2+y^2+z^2)x\big)=3x^2+y^2+z^2\). Adding the three yields \(5(x^2+y^2+z^2)\). By the product rule: \(\nabla f\cdot\mathbf r+f\,\operatorname{div}\mathbf r=2r^2+3r^2\).
Review the concept: Divergence
The divergence of a vector field \(\mathbf F=\langle P,Q,R\rangle\) is the scalar field that measures, at each point, how much the flow of the field expands or compresses.
\[\operatorname{div}\mathbf F=\nabla\cdot\mathbf F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\]
More practice problems: divergence
- ZW-DIVERGENCE-001 Compute divmathbf F for mathbf F=langle x^2, y^2, z^2rangle.
- ZW-DIVERGENCE-002 Compute divmathbf F for mathbf F=langle xy, yz, zxrangle.
- ZW-DIVERGENCE-003 The divergence of mathbf F=langle x, y, zrangle at any point equals:
- ZW-DIVERGENCE-004 Compute divmathbf F for the rotation field mathbf F=langle -y, x, 0rangle.
- ZW-DIVERGENCE-005 Compute divmathbf F for mathbf F=langle xe^y, ye^z, ze^xrangle.
- ZW-DIVERGENCE-006 Evaluate divmathbf F at (1,2,3) for mathbf F=langle x^2y, yz, xz^2rangle.
- ZW-DIVERGENCE-007 If divmathbf F>0 at a point P, then near P the flow of the field:
- ZW-DIVERGENCE-008 Which of these fields is incompressible, that is, has divmathbf F=0?