Question

1) If a vector field A(x,y,z)=xz³e_x - 2x²yze_y+2yz⁴e_z , calculate the divergence ∇·A at point M(1,-1,-1).

Answer 1

Gives

$$A(x,y,z)=xz^3e_x-2x^2yze_y+2yz^4e_z$$

$$\nabla.A=(\hat{e_x} \frac{d}{dx}+\hat{e_y}\frac{d}{dy}+\hat{e_z}\frac{d}{dz})({xz^3e_x-2x^2yze_y+2yz^4e_z})$$

$\nabla.A=z^3-2x^2z+8yz^3$

$\nabla.A|_({1,-1,-1})=9$