Hey can anyone help me with this flux integral problem from multivariable Calculus?
F=xzi+yk S is the hemisphere x^2+y^2+z^2=9 z/=0 oriented upward.

this is what I got so far:
r=(x,y,z)
partial(r/x)=(1,0,-x/sqrt(9-x^2-y^2))
partial(r/y)=(0,1,-y/sqrt(9-x^2-y^2))
cross product=(x/sqrt(9-x^2-y^2), y/sqrt(9-x^2-y^2),1)
let sqrt(9-x^2-y^2)=c
integral F*crossproduct dA=integral(zx^2/c+y^2/c)dydx evaluated from y=sqrt(9-x^2) to y=-sqrt(9-x^2) and x=3 to x=-3. I don't understand how to set up the integral , please help.

Question

Hey can anyone help me with this flux integral problem from multivariable Calculus?
F=xzi+yk S is the hemisphere x^2+y^2+z^2=9 z>/=0 oriented upward.

 this is what I got so far:
r=(x,y,z)
partial(r/x)=(1,0,-x/sqrt(9-x^2-y^2))
partial(r/y)=(0,1,-y/sqrt(9-x^2-y^2))
cross product=(x/sqrt(9-x^2-y^2), y/sqrt(9-x^2-y^2),1)
let sqrt(9-x^2-y^2)=c
integral F*crossproduct dA=integral(zx^2/c+y^2/c)dydx evaluated from                   y=sqrt(9-x^2) to y=-sqrt(9-x^2) and x=3 to       x=-3. I don't understand how to set up the integral , please help.

mathteacher1729 · Answer

There is a hemisphere example here (with illustrations and all steps worked out) PDF http://banach.millersville.edu/~bob/math311/SurfaceIntegrals/main.pdf

anonymous · Answer

I hadno idea on how to set it up but that powerpoint cleared it. Thanks

mathteacher1729 · Answer

Great! :)

anonymous · Answer

Use Stokes theorm

anonymous · Answer

The flow of the field is equal to the work done by the field on the cricle
$ x^2 + y^2 =9$
Your answer should be 0 (zero)