(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the surface S with the graph z = 1-x[itex]^{2}[/itex]-y[itex]^{2}[/itex] with z≥0, and also the unit disc in the xy plane. Give this surface an outer normal. Compute: [itex]\int\int_{S}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex]d[itex]\vec{S}[/itex]

where [itex]\vec{F}[/itex](x,y,z) = (2x,2y,z)

2. Relevant equations

[itex]\int\int_{S}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex]d[itex]\vec{S}[/itex]

= [itex]\int\int_{D}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex]([itex]\vec{T_{x}}[/itex][itex]\times[/itex][itex]\vec{T_{y}}[/itex])dxdy

= [itex]\int\int_{D}[/itex] [ [itex]F_1[/itex](-dg/dx)+[itex]F_2[/itex](-dg/dy)+[itex]F_3[/itex] ] dxdy

Where Tx and Ty are the tangent vectors and D is the domain of the parameterization of the surface of S.

3. The attempt at a solution

I found the integrand of the 3rd relevant equation to be -5x^2 - 5y^2 + 1. But I cannot find the bounds of integration. Is the domain D the unit disc or is it just the rectangle -1<x<1 and -1<y<1? If it's the unit disc, then using polar coordinates my solution is -6*Pi/4. If it's the rectangle, my solution is -28/3.

Also, is this just flux through the open surface z = 1-(x^2)-(y^2) or is it including the bottom cap? If not, would I then have to calculate and add the flux through the disc, which should be relatively easy? Then, would these two pieces combined be equal to the integral of div(F) over the volume enclosed by these surfaces, via the divergence theorem?

Thanks for the help.

EDIT: D'oh! The integrand is actually 3x^2 +3y^2 + 1, and I have a feeling the limits of integration are over the rectangle, so the integral over that is 12.

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# Homework Help: Flux of a vector field through warped sphere

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