I have found this problem in my textbook.
Evaluate $$\iint_S F\cdot n ds $$ where $$F = yz\hat i+xz\hat j+xy\hat k$$ and
S is that part of the surface of the sphere $$x^2+y^2+z^2 = 1$$ which
lies in first octant.
I have tried to solve this problem using $x = rcos\theta \cdot sin\phi $ and $y = rsin\theta \cdot sin\phi$
. But i'm stuck in determining the limit. In my text book they used $\phi : 0\to\pi/2$ and $\theta : 0\to\pi/2$ . But i couldn't understand it. Can anyone please explain it? Thanks in advance.
Best Answer
The reason why $\phi$, the azimuthal angle only goes from $0$ to $\frac{\pi}{2}$ is because we only take a quarter-turn along the circle in the $xy$-plane. Since there are $2\pi$ radians in a circle and we are only taking a quarter turn, this reduces to $\frac{1}{4}2\pi = \frac{\pi}{2}$. As for $\theta$, the polar angle, the range of this from the $+z$-axis to the $-$z-axis is $\pi$ radians. Because we're only interested in the first octant, or the part where $z>0$, then we only go half the total distance (we arc from the $+z$-axis down to the $xy$-plane), and so we only $\frac{\pi}{2}$ radians for $\theta$.