Multivariable Calculus question?
If you could, please help solve the problem below by showing the steps to find the solution.
Use the Divergence Theorem to evaluate ∫∫S (9x + 3y + z2)dS where S is the sphere x2 + y2 + z2 = 1.
I previously got the answers 8pi and 16pi but neither are correct.
Update:Just to be clear that is ∫∫S (9x + 3y + z^2)dS and x^2 + y^2 + z^2 = 1
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Note that the normal to S is given by...
Since z = ±√(1 - x^2 - y^2), we have normal vector <-z_x, -z_y, 1>
= <±x/√(1 - x^2 - y^2), ±y/√(1 - x^2 - y^2), 1>.
= <x/z, y/z, 1>.
This has length √((x/z)^2 + (y/z)^2 + 1) = √((x^2 + y^2 + z^2)/z^2) = 1/z, since we are on the unit sphere. So, a unit vector in the direction of n is <x, y, z>.
So, ∫∫s (9x + 3y + z^2) dS
= ∫∫s <9, 3, z> · <x, y, z> dS
= ∫∫s <9, 3, z> · dS (with dS now as a vector)
Now, we can apply the Divergence Theorem, obtaining
∫∫∫ div<9, 3, z> dV
= ∫∫∫ 1 dV
= 4π/3, the volume of the unit sphere.
I hope this helps!
The progression is going as such (from my information): Pre-Calculus, Calc AB (Calc I) Calc BC (Calc II) Multi-variable Calc (Calc III) ...then, Vector diagnosis or some thing comparable to that.