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Math Help - Vector Calculus

  1. #1
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    Vector Calculus

    A solid is defined by a surface:
    x^2 + y^2 + z^2 ≤ b^2

    Whereby, 0 ≤ x, 0 ≤ y, 0≤ z

    The density varies in proportion to the distance from the centre.

    Hence deduce the mass of the solid.


    I assume i use sphereical polar coordinates, but not entirely sure where i go beyond that.

    Any help is appreciated!
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  2. #2
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    In general, if you want to compute the mass of an object occupying volume V with density \rho, then you must compute the integral

    M=\iiint_{V}\rho\,dV.

    Spherical coordinates are definitely the way to go, although you could probably work with cylindrical as well. As I see it, you need to assemble three pieces: the limits for each integral, the density, and the volume differential. What are each of those?
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  3. #3
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    You could calculate \int_0^bp(r)S(r)\,dr where p(r) is the density at distance r from the center and S(r) is 1/8 of the area of a sphere of radius r.
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  4. #4
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    ok the method i've gone for (which i've been told works)

    M = \iiint \rho (r) f(r,\varphi, \theta) |J| dr d\varphi d\theta<br />
    Where J is the Jacobian (which i deduced to be {-r}^{2 }sin(\theta)

    Note \rho (r) = rk where k is a constant

    For some reason (which i do not fully understand), f(r,\varphi, \theta) = 1. Hence the integral simplifies.

    However, im struggling to compute the limits of integration. I know 0 ≤ x, 0 ≤ y, 0≤ z
    So we're concentrating in the first octant, but how do i deduce the integrals?
    Is it just r between 0 and r, \varphi between 0 and \frac{\pi }{2 } and the same for \theta ? Or is r = {a}

    Hope you can clear my confusion
    Last edited by imagemania; May 18th 2011 at 06:49 AM.
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  5. #5
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    It would seem \theta has limits \pi ...0

    Which gets me what seems to be a reasonable answer of:
    \frac{k{a}^{4}\pi}{2}

    Can anyone explain this, if this is correct, i dno not know how to choose the correct limits
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  6. #6
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    The first octant, which is the octant over which you're integrating your sphere, has limits of the polar angle \theta\in[0,\pi/2], and the azimuthal angle \varphi\in[0,\pi/2].

    I don't think your Jacobian is correct. The element of volume is positive, not negative. You have the correct magnitude.
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