Answer to Question #313953 in Math for Jyo

Question

Compute the divergence and curl of the vector point functions. 𝐹 = 𝑋 ^2𝑌𝑍𝑖 − 2𝑋 𝑍^ 3 𝑗 + 𝑋 𝑍^ 2𝑘.

Answer 1

$divF=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}=2xyz-0+2xz=2xz\left( y+1 \right) \\rotF=\left| \begin{matrix} i& j& k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ x^2yz& -2xz^3& xz^2\\\end{matrix} \right|=i\left( 0+6xz^2 \right) -j\left( z^2-x^2y \right) +k\left( -2z^3-x^2z \right) =\\=\left( 6xz^2,x^2y-z^2,-x^2z-2z^3 \right)$