Question

Evaluate both sides of the divergence theorem for the vector field A= 5xy ax + xy² ay + 4z az
defined in the region 1≤ x ≤ 3, −2 ≤ y ≤ 4 and −1≤ z ≤ 2 .

Answer 1

Solution

Vector field

$A=5xya_x+xy^2 a_y+4za_z$

Del operator

$\nabla=a_x \frac{d}{dx}+a_y \frac{d}{dy}+a_z\frac{d}{dz}$

Now divergence is written as

$\nabla.A=\frac{d(5xy)}{dx}+\frac{d(xy^2)}{dy}+\frac{d(4z)}{dz}$

$\nabla.A=5y+2x+4$

Now using divergence theorem

$\intop \nabla . A dV=\iiint \nabla.A dx dy dz$

=$\int^2_{-1}\int^4_{-2} \intop^3_1(5y+2x+4) dx dy dz$

So finally we get

$\intop \nabla . A dV=408$