Gauss Divergence Theorem

Q. The value of the integral $∯\overrightarrow{r.}\overrightarrow{n}dS$ over the closed surface S bounding a volume V, where $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is the position vector and $\overrightarrow{n}$ is the normal to the surface S, is

1. V
2. 2V
3. 3V
4. 4V

Q. The Gauss divergence theorem relates certain

1. surface integrals to volume integrals
2. surface integrals to line integrals
3. vector quantities to other vector quantities
4. line integrals to volume integrals

Q. Given a vector field F, the divergence theorem states that

1. ${\oint }_s\overrightarrow{F}.d\overline{S}={\oint }_V\nabla .\overrightarrow{F}dv$
2. ${\oint }_s\overrightarrow{F}.d\overrightarrow{S}={\oint }_V\nabla \times \overrightarrow{F}dv$
3. ${\oint }_s\overrightarrow{F}.d\overrightarrow{S}={\oint }_V\left(\nabla \overrightarrow{F}\right)dv$
4. ${\oint }_s\overrightarrow{F}.d\overline{S}={\oint }_V(\nabla \times \overrightarrow{F})dv$

Q.

The volume of the solid generated by revolving about the $y$ axis bounded by the parabola $y=x^2$and $x=y^2$ is

1. $\frac{2}{15}\mathrm{\pi }$

2. $\frac{24}{5}\mathrm{\pi }$

3. $\frac{3}{10}\mathrm{\pi }$

4. $\frac{5}{24}\mathrm{\pi }$

Q. A vector field D =$2{\rho }^2{a}_{\rho }+z{a}_z$ exists inside a cylindrical region enclosed by the surfaces $\rho$ = 1, z = 0 and z = 5. Let S be the surface bounding this cylindrical region. The surface integral of this field on S ($∯D.ds$) is_______

1. 78.53