Exercise · ZW-DIVERGENCE-005 · ★★☆ Intermediate
Compute \(\operatorname{div}\mathbf F\) for \(\mathbf F=\langle xe^{y},\,ye^{z},\,ze^{x}\rangle\).
Choose one option.
Show answer
- B. \(e^{x}+e^{y}+e^{z}\)
Explanation
Differentiate each component with respect to its own variable: \(\partial_x(xe^{y})=e^{y}\), \(\partial_y(ye^{z})=e^{z}\), \(\partial_z(ze^{x})=e^{x}\). Cross derivatives play no role.
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-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?
- ZW-DIVERGENCE-009 For which value of a does mathbf F=langle ax, 3y, -2zrangle have zero divergence?