GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Hemisphere $\left({\mathit{x}}^2+{\mathit{y}}^2+{\mathit{z}}^2=1;\mathit{z}\ge 0\right)$
To find the integration of the given expiration, it is easy if we are using the stoke's theorem. Surface integral will become line integral.
$\int \left(\nabla \times \overrightarrow{\mathit{u}}\right)\cdot \stackrel{\wedge }{\mathit{n}}\mathit{d}\mathit{S}=\int \overrightarrow{\mathit{u}}\cdot \mathit{d}\mathit{r}$
Now, putting the value of $\mathit{u}$ in the above equation.$={\oint }_{\mathit{c}}\frac{1}{3}\left(-{\mathit{y}}^3\mathit{d}\mathit{x}+{\mathit{x}}^3\mathit{d}\mathit{y}\right)\because \mathit{Z}=0$ at base circle
$={\oint }_{\mathit{c}}\left({\mathit{y}}^2\mathit{d}\mathit{x}\mathit{d}\mathit{y}+{\mathit{x}}^2\mathit{d}\mathit{x}\mathit{d}\mathit{y}\right)$ [using green theorem]
$=\int \int \left({\mathit{y}}^2+{\mathit{x}}^2\right)\mathit{d}\mathit{x}\mathit{d}\mathit{y}$
Now converting the above equation into polar coordinate
${\mathit{x}}^2+{\mathit{y}}^2={\mathit{r}}^2$