Gauss divergence theorem:

• It states that the surface integral of the normal component of a vector function \(\vec F\) taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function \(\vec F\) taken over a volume enclosed by the closed surface ‘S’.
• Mathematically, it can be written as:
  \(\Rightarrow \underset{S}{\mathop \iint }\,\vec{F}.\hat{n}ds=\iiint\limits_{V}{\nabla .\vec{F}dv}\)

Calculation:

Given:
The surface of the given sphere is S = x2 + y2 + z2 = 9, it is a closed surface, \({\rm{\bar F\;}} = {\rm{\;}}9{\rm{x\bar i\;}} - {\rm{\;}}2{\rm{y\bar j\;}} - {\rm{\;z\bar k}}\)

\( \underset{S}{\mathop \iint }\,\vec{F}.\hat{n}ds=\iiint\limits_{V}{\nabla .\vec{F}dv}\)

\(\iint_{s}^{}\left (9x\vec{i}\ -\ 2y\vec{j}\ -\ z\vec{k} \right )\ .\ \vec{n}\ ds\) = \(\iiint\limits_{V}{(\frac{\delta}{\delta x}(9x)\ +\ \frac{\delta}{\delta y}(- 2y)\ +\ \frac{\delta}{\delta z}(- z)dx \ dy\ dz}\)

\(\iint_{s}^{}\left (9x\vec{i}\ -\ 2y\vec{j}\ -\ z\vec{k} \right )\ .\ \vec{n}\ ds\) =  \(\iiint\limits_{V}{(9\ -\ 2\ -\ 1)dx\ dy\ dz}\)

\(\iint_{s}^{}\left (9x\vec{i}\ -\ 2y\vec{j}\ -\ z\vec{k} \right )\ .\ \vec{n}\ ds\) = \(6\left ( \iiint_{V}^{}1\ dx\ dy\ dz \right )\)

\(\iint_{s}^{}\left (9x\vec{i}\ -\ 2y\vec{j}\ -\ z\vec{k} \right )\ .\ \vec{n}\ ds\) = 6\((\frac{4π}{3}r^3)_{r=3}\)

\(\iint_{s}^{}\left (9x\vec{i}\ -\ 2y\vec{j}\ -\ z\vec{k} \right )\ .\ \vec{n}\ ds\) = 6\((4\frac{π}{3}(3)^3)\) = 216π = 678.580