Curl and Divergence — Question 8

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.