Gauss Divergence Theorem Engineering Maths, Btech first year

Gauss's Divergence Theorem

This theorem gives relation between Surface Integral and Volume Integral.
The surface integral of the normal component of a vector function $\overrightarrow{F}$ taken around a closed surface $S$ is equal to the integral of the divergence of $\overrightarrow{F}$ taken over volume $V$ enclosed by the surface $S$. $${\iint }_S\overrightarrow{F}.\hat{N}ds={\iiint }_V\overrightarrow{F}dV$$

Sample Problem

Usinf Gauss's Divergence Theorem, evaluate ${\iint }_S\overrightarrow{F}.\hat{N}ds$ if equation of the sphere $S$ is $x^2+y^2+z^2=16$ and $\overrightarrow{F}=3x\hat{i}+4y\hat{j}+5z\hat{k}$.

Solution

By Gauss's Divergence Theorem, $${\iint }_S\overrightarrow{F}.\hat{N}ds={\iiint }_V\overrightarrow{F}dV$$ So calculating divergence, $$\mathrm{\nabla }.\overrightarrow{F}=(\hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}).(3x\hat{i}+4y\hat{j}+5z\hat{k})$$ $$=3+4+5=14$$ Using the value of $\mathrm{\nabla }\overrightarrow{F}$ $${\iint }_S\overrightarrow{F}.\hat{N}ds={\iint }_V\int 14.dv$$ $$=14V$$ Since $V$ is the volume of sphere, $$V=\frac{4}{3}\pi r^3$$ $$\text{So }14V=14\times \frac{4}{3}\pi (4{)}^3=\frac{3584\pi }{3}$$ where 4 is the radius of the given sphere.