# VC5.24

Here $F=zi+xj+yk\,$ --(1)

Also,$r=xi+yj+zk,dr=dxi+dyj+dzk\,$

$F\cdot dr=zdx+xdy+ydz\,$

Therefore,$\oint _{C}F\cdot dr=\oint _{C}(zdx+xdy+ydz)\,$ --(2)

On the circle C, $x^{2}+y^{2}=1,z=0\,$ on the xy-plane. Hence on C,we have z=0 so that dz=0.Hence,from (2)reduces to $\oint _{C}F\cdot dr=\oint _{C}xdy\,$ --(3)

Now the parametric equations of C are $x=\cos \theta ,y=\sin \theta \,$ --(4)

Using (4),(3) reduces to $\oint _{C}F\cdot dr=\int _{{0}}^{{2\pi }}\cos \theta \cos \theta d\theta \,$

=$\int _{{0}}^{{2\pi }}{\frac {1+\cos 2\theta }{2}}d\theta ={\frac {1}{2}}[\theta +{\frac {\sin 2\theta }{2}}]_{{0}}^{{2\pi }}=\pi \,$ --(5)

Let P(x,y,z) be any point on the surface of the hemisphere $x^{2}+y^{2}+z^{2}=1\,$. The spherical polar coordinates of P are given by $x=\sin \phi \cos \theta ,y=\sin \phi \sin \theta ,z=\cos \phi \,$ --(6)

Here $x^{2}+y^{2}+z^{2}=1\,$ hence the direction cosines of OP are in $\sin \phi \cos \theta ,\sin \phi \sin \theta ,\cos \theta \,$.But OP is also the direction of outward drawn normal at P.

Therefore,n=unit normal vector to hemisphere S=$\sin \theta \cos \phi i+\sin \theta \sin \phi j+\cos \theta k\,$ --(7)

Also,${\mathrm {curl}}F={\begin{vmatrix}i&j&k\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\z&x&y\end{vmatrix}}\,$

=$i+j+k\,$ on simplification. --(8)

Therefore,${\mathrm {curl}}F\cdot n=\sin \theta \cos \phi +\sin \theta \sin \phi +\cos \theta \,$ from (7) and (8) --(9)

Therefore,$\iint _{S}{\mathrm {curl}}F\cdot ndS\,$

=$\int _{{\phi =0}}^{{{\frac {\pi }{2}}}}\int _{{\theta =0}}^{{2\pi }}(\sin \theta \cos \phi +\sin \theta \sin \phi +\cos \theta )\sin \theta d\theta d\phi \,$

=$\int _{{\phi =0}}^{{{\frac {\pi }{2}}}}{\sin ^{2}\phi [\sin \theta ]_{{0}}^{{2\pi }}+\sin ^{2}\phi [-\cos \theta ]_{{0}}^{{2\pi }}+\sin \phi \cos \phi [\theta ]_{{0}}^{{2\pi }}}d\phi \,$

=$\int _{{\phi =0}}^{{{\frac {\pi }{2}}}}(0+0+2\pi \sin \phi \cos \phi )d\theta =\pi \int _{{\phi =0}}^{{{\frac {\pi }{2}}}}\sin 2\phi d\phi =\pi [-{\frac {\cos 2\phi }{2}}]_{{0}}^{{{\frac {\pi }{2}}}}=-{\frac {\pi }{2}}[-1-1]=\pi \,$ --(10)

From (5) and (10),Stokes'theorm is verified.