Question

Compute the flux of the vector field, F, through the surface, S.
vector F= x i + y j + z k and S is the sphere x^2 + y^2 + z^2 = a^2 oriented outward.

Answer 1

@eliassaab can you help me with this one

Answer 2

@perl @dan815

Answer 3

I keep getting zero

Answer 4

@ganeshie8

Answer 5

A parametrization of the sphere is
\[(a \cos (\theta ) \sin (\phi ),a \sin
(\theta ) \sin (\phi ),a \cos (\phi )),\quad 0\le \theta \le 2 \pi ; \quad
0\le \phi \le \pi\]

Answer 6

The normal vector is
\[\left(a^2 \cos (\theta ) \sin ^2(\phi ),a^2
\sin (\theta ) \sin ^2(\phi ),a^2 \sin
(\phi ) \cos (\phi )\right)\]

Answer 7

Dot multiply your field with the normal vector, you get
\[a^3 \sin (\phi )
\]

Answer 8

So the flux is

\[\int _0^{\pi }\int _0^{2 \pi }a^3 \sin (\phi
)d\theta d\phi =4 \pi a^3\]

Answer 9

why is it from 0 to pi and not 0 to pi/2

Answer 10

@eliassaab thank you by the way

Answer 11

YW. From 0 to pi/2 you get the semi-sphere above the xy-plane