Divergence

Given vector, $\vec{A} = x^2 z \hat{i} - 2y^3y^2 \hat{j} +xy^2z \hat{k}$ and alternative point $(1,-1,1)$ .

The divergence of $\vec{A} = \vec{\nabla} \cdot \vec{A}$

$= \left( \hat{i} \dfrac{∂}{∂x} + \hat{j} \dfrac{∂}{∂y} + \hat{k} \dfrac{∂}{∂z} \right) \cdot \left(x^2z \hat{i} - 2y^3z^2 \hat{j} + xy^2 z \hat{k}\right)$

$= \dfrac{∂}{∂x} (x^2z) + \dfrac{∂}{∂y} (-2y^3z^2) +\dfrac{∂}{∂z} (xy^2z)$

$=2xz - 6y^2z^2 + xy^2$

Therefore, the divergence at point $(1,-1,1):$

$=2\times 1 \times 1 - 6(-1)^2 (1)^2 + 1 \times (-1)^2$

$=2-6+1 = -3$