Hi,

I am working on the surface integral - flux section. I feel so desperate. Looking at the book for two days,still can't conclude how to solve this kind of questions.

I don't quiet get the idea of this section for how to compute the flux. It seems that we need to memories many formulas in order to solve flux question given in the form of z=g(x,y), y=g(x,z) or x= g(y,z).

I am just wondering can anyone give me a simple conclusion or lecture of how to solve flux questions.

question 1.
Evaluate the flux of the vector field F(x,y, z) = -yi + xj -zk
through the unit sphere x^2 + y^2 +z^2 =1 that has downward orientation.

question 2
Evaluate the flux of the vector field F (x,y,z) = <x, -1, z> through the surface S that has downward orientation and given by the equation z=xcosy where 0<or = x < or =1, pi/4 < or = y < or = pi/3.

2. Originally Posted by kittycat
question 1.
Evaluate the flux of the vector field F(x,y, z) = -yi + xj -zk
through the unit sphere x^2 + y^2 +z^2 =1 that has downward orientation.
I might be a n00b in physics but I am quite sure the flux is, the surface integral:
$\iint_S \bold{F}\cdot \bold{n}$
Where $S$ is the surface, $\bold{F}$ is the vector field and $\bold{n}$ is normal vector.

The first step is to parametrize the surface. That can be written as $\bold{G}(\theta,\phi) = (\sin \phi \cos \theta, \sin \phi \sin \theta, \cos \phi)$ for $0\leq \phi,\theta \leq 2\pi$.

Next is you compute the partials,
$\partial_{\theta}\bold{G} = (-\sin \phi \sin \theta, \sin \phi \cos \theta, 0)$
$\partial_{\phi}\bold{G} = (\cos \phi \cos \theta, \cos \phi \sin \theta, - \sin \phi)$.

Now compute,
$\partial_{\theta}\bold{G} \times \partial_{\phi}\bold{G}= (-\sin^2\phi \cos \theta, \sin^2\phi \sin\theta, -\sin \phi \cos\phi \sin^2\theta - \sin\phi \cos \phi \cos^2 \theta)$ $=(-\sin^2\phi \cos \theta, \sin^2\phi \sin\theta,-\sin\phi\cos\phi)$

But I could have also computed $\partial_{\phi}\bold{G}\times \partial_{\theta}\bold{G}$ and got the same vector only with a negative. So which way do I pick it? Remember the problem says the normal vector $\bold{n}$ points downward. So pick a point on the sphere, say, $(1,0,0)$ the vector should be $k(-1,0,0)$ for $k>0$, i.e. a vector pointing some positive scalar multiple in the same direction as its inward normal $-\bold{i}$. Is that true? Let us just check this, on this point $\theta = 0,\phi = \pi/2$, and it is true! So we have the inward normal vector parametrization.

Thus, the surface integral is,
$\iint_R \bold{F}((-\sin^2\phi \cos \theta, \sin^2\phi \sin\theta,-\sin\phi\cos\phi))\cdot$ $((-\sin^2\phi \cos \theta, \sin^2\phi \sin\theta,-\sin\phi\cos\phi)) dA$

$\iint_R (-\sin^2\phi \sin\theta, -\sin^2\phi \cos\theta, \sin\phi\cos\phi)\cdot$ $(-\sin^2\phi \cos \theta, \sin^2\phi \sin\theta,-\sin\phi\cos\phi)dA$
Which simplifies to,
$\int_0^{2\pi} \int_0^{2\pi} -2\sin\phi \cos\phi d\theta \phi =0$