1. ## surface integrals

Consider the hemispherical volume V defined by:
x^2 + y^2 + z^2 =< a^2, z>=0, and denote its closed surface by S.
Define the vector field f=(-y, x, z^2)

a) Evaluate the net outward flux of f through S by direct integration.

{to parameterise the curved surface use spherical polar coordinates}

plz help me its due in a few hours

2. Originally Posted by martinr
Consider the hemispherical volume V defined by:
x^2 + y^2 + z^2 =< a^2, z>=0, and denote its closed surface by S.
Define the vector field f=(-y, x, z^2)

a) Evaluate the net outward flux of f through S by direct integration.

{to parameterise the curved surface use spherical polar coordinates}

plz help me its due in a few hours
The vector-function $\bold{G}: U\mapsto \mathbb{R}^3$ defined by $\bold{G}(\theta , \phi) = (a\sin \phi \cos \theta , a\sin \phi \sin \theta , a\cos \theta)$ where $U = [0,2\pi]\times [0,\pi/2)$ will parametrize the upper hemi-sphere.

Then, $\int_V \bold{F} \cdot d\bold{S} = \pm \int_U (-y,x,z^2) \cdot \left( \partial_{\theta} \bold{G}\times \partial_{\phi} \bold{G} \right)$.
Where $\pm$ depends whether the parametrization was inward or outward. Just pick a vector and check which was it goes.