Question

Show the complete solution on the following questions.

1. Evaluate $\nabla \cdot (F \times G),$ given $𝐹(𝑥,𝑦,𝑧) = 𝑦𝑧 \vec{𝑖} + 𝑥𝑧\vec{ 𝑗} + 𝑥𝑦\vec{ 𝑘}$ and $𝐺(𝑥,𝑦,𝑧) = 𝑥𝑦 \vec{𝑗} + 𝑥𝑦𝑧\vec{ 𝑘}$

Answer 1

$F×G=(F_yG_z-F_zG_y)\vec{i}+(F_zG_x-F_xG_z)\vec{j}+(F_xG_y-F_yG_x)\vec{k}$

$F×G=(x^2yz^2-x^2y^2)\vec{i}+(-xy^2z^2)\vec{j}+(xy^2z)\vec{k}$

$\nabla⋅(F×G)={\frac \partial {\partial x}}(x^2yz^2-x^2y^2)+{\frac \partial {\partial y}}(-xy^2z^2)+{\frac \partial {\partial z}}(xy^2z)$

$\nabla⋅(F×G)=2xyz^2-2xy^2-2xyz^2+xy^2=-xy^2$