Curl and Divergence — Question 8

Question 8

A linear planar field has the expansion–rotation form $$\mathbf F(x,y)=\langle ax-by,\,bx+ay\rangle.$$ Its divergence is $4$ and its scalar curl is $6$ everywhere.

Tasks

1. Determine $a$ and $b$.

2. Write the resulting field explicitly.

3. Separate it into pure expansion and pure rotation parts.

Question 8 – Solution

Strategy. Divergence isolates the expansion coefficient, while scalar curl isolates twice the rotation coefficient.

Step 1: Compute the invariants $$\nabla\cdot\mathbf F=a+a=2a, \qquad \operatorname{curl}_{2D}\mathbf F=b-(-b)=2b.$$ Thus $$2a=4, \qquad 2b=6,$$ so $\boxed{a=2}$ and $\boxed{b=3}$.

Step 2: Write the field $$\boxed{\mathbf F(x,y)=\langle 2x-3y,\,3x+2y\rangle}.$$

Step 3: Decompose $$\mathbf F =\underbrace{\langle 2x,2y\rangle}_{\text{pure expansion}} +\underbrace{\langle-3y,3x\rangle}_{\text{counterclockwise rotation}}.$$ The first part has divergence $4$ and zero curl; the second has zero divergence and scalar curl $6$.

Verification Recombining the two displayed parts returns both components of the boxed field and reproduces the prescribed invariants.