Calculus Question: help appreciated.

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Hydralisk
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Calculus Question: help appreciated.

Postby Hydralisk » Thu Dec 11, 2008 2:42 pm UTC

Quick Calculus Question here.It's 2nd Year uni work so don't panic if you can't get it! :p

A solid of variable density is in the form of a sphere (x^2 + y^2 + z^2 < a^2, where a>0. The density at the point (x,y,z) is z^2. Use triple integration to find the mass of the solid.

Solution given:
Spoiler:
δΦ/δn = n.gradΦ where n=1/9(1,8,-4)

Hence, grad Φ = (y^2, 2xyz-5, xy^2) = (2,7,3) at (3,1,2).

So δΦ/δn = 1/9(1,8,-4).(2,7,3) = 1/9(2+56-12) = 46/9


I understand most of the question (basically from line 2 down), but what I can't work is where the 1/9th came from. Any help please? :l

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qinwamascot
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Re: Calculus Question: help appreciated.

Postby qinwamascot » Thu Dec 11, 2008 3:29 pm UTC

Did you try spherical coordinates? It's a fairly easy integral if I'm doing it right after converting.

edit: didn't read your spoiler. I'm checking it right now.
edit2: I really don't understand what's going on here. I'd just convert the integral into spherical coordinates and integrate the density function to get the total mass; this doesn't seem to be what's being done in the solution and I couldn't figure out what was being done.
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boxcounter
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Re: Calculus Question: help appreciated.

Postby boxcounter » Thu Dec 11, 2008 4:10 pm UTC

The "spoiler" is actually an answer to an entirely different question -- probably one on Lagrange optimization.

Ended
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Re: Calculus Question: help appreciated.

Postby Ended » Thu Dec 11, 2008 4:19 pm UTC

Hydralisk wrote:what I can't work is where the 1/9th came from. Any help please? :l

The 1/9 is just a normalization factor, needed to make ||n|| = 1. But as others have said, the spoilered answer doesn't match the question.

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Generic Protoplasm
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Re: Calculus Question: help appreciated.

Postby Generic Protoplasm » Sat Dec 13, 2008 11:51 pm UTC

Convert to spherical coordinates and integrate the density function (after converting it to spherical also).
Just remember that [imath]dx dy dz = \rho^2 sin \phi d\rho d\phi d\theta[/imath]
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