Curl and Divergence — Question 4

Question 4

For constants $a,b,d$, consider $$\mathbf F(x,y,z)=\langle ax-by,\,bx-ay,\,dz\rangle.$$ Classify all parameter triples $(a,b,d)$ for which the field is both divergence-free and curl-free on $\mathbb R^3$.

Tasks

1. Compute the divergence.

2. Compute the curl.

3. Solve the simultaneous parameter conditions and verify the resulting family.

Question 4 – Solution

Strategy. Treat the scalar divergence and all three curl components as simultaneous identities.

Step 1: Divergence $$\nabla\cdot\mathbf F=a+(-a)+d=d.$$ Thus divergence-free requires $d=0$.

Step 2: Curl With $P=ax-by$, $Q=bx-ay$, and $R=dz$, $$\nabla\times\mathbf F =\langle 0,0,Q_x-P_y\rangle =\langle 0,0,b-(-b)\rangle =\langle 0,0,2b\rangle.$$ Thus curl-free requires $b=0$.

Step 3: Classify The parameter $a$ is unrestricted, so $$\boxed{(a,b,d)=(a,0,0),\quad a\in\mathbb R}.$$ The resulting field is $$\mathbf F=\langle ax,-ay,0\rangle.$$

Verification Its divergence is $a-a+0=0$, and all cross derivatives vanish. Hence every field in the boxed family satisfies both conditions, including the zero field when $a=0$.