Curl and Divergence — Question 10

Question 10

Let $$\mathbf F(x,y,z)=\langle-yz,\,xz,\,xy\rangle.$$

Tasks

1. Compute the divergence and curl everywhere.

2. Evaluate the curl at $P=(1,2,3)$ and give its direction.

3. Find every point at which the curl vanishes.

Question 10 – Solution

Strategy. Compute the global formulas first; then solve the component equations of the curl simultaneously.

Step 1: Divergence Each component omits its matching differentiation variable, so $$\nabla\cdot\mathbf F =\frac{\partial(-yz)}{\partial x} +\frac{\partial(xz)}{\partial y} +\frac{\partial(xy)}{\partial z} =\boxed{0}.$$

Step 2: Curl $$\begin{align*} \nabla\times\mathbf F &=\langle R_y-Q_z,\,P_z-R_x,\,Q_x-P_y\rangle\\ &=\langle x-x,\,-y-y,\,z-(-z)\rangle\\ &=\boxed{\langle 0,-2y,2z\rangle}. \end{align*}$$ At $P=(1,2,3)$, $$(\nabla\times\mathbf F)(P)=\boxed{\langle 0,-4,6\rangle},$$ which points in the direction $\boxed{\langle 0,-2,3\rangle/\sqrt{13}}$.

Step 3: Vanishing locus The curl is zero exactly when $$-2y=0, \qquad 2z=0.$$ Thus $y=z=0$ while $x$ is arbitrary: $$\boxed{\nabla\times\mathbf F=\mathbf 0\text{ exactly on the }x\text{-axis}}.$$

Verification Substituting $(x,0,0)$ into the curl formula gives $\mathbf 0$, and any point off the $x$-axis has $y\ne 0$ or $z\ne 0$, making at least one curl component nonzero.