Solution:

$\bar A = x^{2} z \bar i -2y^{2} z^{2} \bar j + xy^{2} z \bar k$

The divergence of $\bar A$ is given by

$\nabla. \bar A = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$

$\begin{aligned} \frac{\partial A_x}{\partial x} &= 2xz \\ \frac{\partial A_y}{\partial y} &= -4yz^2 \\ \frac{\partial A_z}{\partial z} &= xy^2 \end{aligned}$

Then, $\nabla . \bar A = 2xz - 4yz^2 + xy^2$

At (1, -1, 1), $\nabla . \bar A = 2(1)(1) - 4(-1)(1)^2 + (1)(-1)^2 = 7$