Question

Please explain clearly and show all steps. Thank you.

A cuboid is bounded by the planes x=0, x=1, y=0, y=3, z=0 and z=2. Use Gauss' Divergence Theorem to calculate SSsF. NºdS, the flux of the vector field F =x2i® + zjº+yk outward of the cuboid through its surfaces.

Answer 1

We know Diveregence theorem is:-

$\int \int_S \boldsymbol{\mathbf{F}}.\mathbf{N}ds=\int \int\int_V divF dV\\\\ Here \; \boldsymbol{F}= x^2i+zj+yk\Rightarrow div F = \bigtriangleup .F=\left( \frac{\partial }{\partial x}i +\frac{\partial }{\partial x}j+ \frac{\partial }{\partial x}k \right).\left ( x^2i+zj+yk \right )\\\\ = 2x+0+0 = 2x$

$\int \int_S \boldsymbol{\mathbf{F}}.\mathbf{N}ds=\int \int\int_V 2x+2 \, dV\\\\ limit \; of\; x\; is\;\; 0\leq x\leq 1\\\\ limit \; of\; y\; is\;\; 0\leq x\leq 3\\\\limit \; of\; z\; is\;\; 0\leq x\leq 2\\\\ taking \; dV = dxdydz So, \int \int\int_V 2x \, dV= \int_0^2 \int_0^3 \int_0^1 (2x)dxdydx\\\\ Or, \int_0^2 \int_0^3 \left [x^2\right ]_0^1\; dydx= \int_0^2 \int_0^31\,dydz\\\\ \int_0^2 \left [ y \right ]_0^3\, dz =\int_0^2 3\,dz \\\\ Or, 3 \int_0^2 dz = 3\times 2=6 \;\;\; Answer$