Curl and Divergence — Question 1

Question 1

For $$\mathbf F(x,y,z)=\langle x^2y,\,yz^2,\,xz\rangle,$$ compute $\nabla\cdot\mathbf F$ and $\nabla\times\mathbf F$.

Tasks

1. Compute the divergence term by term.

2. Compute all three components of the curl.

3. Evaluate both quantities at $(1,2,-1)$.

Question 1 – Solution

Strategy. Label the components $P,Q,R$ and apply the divergence and curl formulas in a fixed order.

Step 1: Divergence With $P=x^2y$, $Q=yz^2$, and $R=xz$, $$\nabla\cdot\mathbf F=P_x+Q_y+R_z =2xy+z^2+x.$$ Thus $$\boxed{\nabla\cdot\mathbf F=2xy+z^2+x}.$$

Step 2: Curl $$\begin{align*} \nabla\times\mathbf F &=\langle R_y-Q_z,\,P_z-R_x,\,Q_x-P_y\rangle\\ &=\langle 0-2yz,\,0-z,\,0-x^2\rangle. \end{align*}$$ Therefore $$\boxed{\nabla\times\mathbf F=\langle-2yz,-z,-x^2\rangle}.$$

Step 3: Evaluate At $(1,2,-1)$, $$\boxed{\nabla\cdot\mathbf F=6}, \qquad \boxed{\nabla\times\mathbf F=\langle 4,1,-1\rangle}.$$

Verification The curl components follow the cyclic order $(R_y-Q_z,P_z-R_x,Q_x-P_y)$; recomputing from the determinant form gives the same signs.