Question

4. Let n be the outward unit normal vector field of the surface Sirr, ) - (rcos)i + (r sin 8j+ (1-)k, 0 STS 1,0 SO S 27, and let F =(x2)i + (ry)j + (yz)k. Evaluate the flux of the Curl /vx Fondo

Answer 1

By Stokes theorem, we have

// xFn do = F. dr JJS

where is the boundary of . Since has parametrization ${\bf r}(r,\theta)=(r\cos\theta,r\sin\theta,1-r)$ , its boundary is

$\{(\cos\theta,\sin\theta,0):\theta\in [0,2\pi]\}$

Thus,

$\begin{align*}\iint_S\nabla\times {\bf F}\cdot{\bf n}~d\sigma&=\int_C{\bf F}\cdot d{\bf r}\\ &=\int_0^{2\pi}(0,\cos\theta\cdot\sin\theta,0)\cdot (-\sin\theta,\cos\theta,0)~d\theta \\ &=\int_0^{2\pi}\cos^2\theta\cdot\sin\theta~d\theta\end{align*}$

Let $\cos \theta=w$ so that $-\sin \theta~d\theta=dw$ ; then

$\begin{align*}\int \cos^2\theta\cdot\sin\theta~d\theta&=-\int w^2~dw\\ &=-\frac {w^3}3\\ &=-\frac 13\cos^3\theta\end{align*}$

Thus,

$\begin{align*}\iint_S\nabla\times {\bf F}\cdot{\bf n}~d\sigma&=\int_C{\bf F}\cdot d{\bf r}\\ &=\int_0^{2\pi}\cos^2\theta\cdot\sin\theta~d\theta\\ &=-\frac 13\cos^3\theta\mid_0^{2\pi}\\ &=0\end{align*}$