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Integration over an Ellipsoidal Domain - Clarification

  1. Aug 7, 2009 #1
    1. The problem statement, all variables and given/known data

    I want to integrate a function over an Ellipsoidal domain.

    [tex]\[
    \underset{\left(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}-1\right)}{\intop\intop\intop}f\left(x,y,z\right)dxdydz\][/tex]


    https://www.physicsforums.com/showthread.php?t=110799&highlight=volume+ellipsoid"
    I have already looked into the above thread and posts descibing this and i found that a bit too difficult to understand hence i tried out a solution of my own.


    2.Question - A

    Please could you tell me if my solution is right.

    3. The attempt at a solution

    I stay in the cartesian coordinate system but make a transoformation of variables and hence i also find the jacobian determinant for the volume transformation.

    [tex]\[
    x=ua,y=vb,z=wc\][/tex]

    and hence i get,

    [tex]
    \[
    \underset{\left(u^{2}+v^{2}+w^{2}-1\right)}{\intop\intop\intop}f\left(ua,vb,wc\right)\left[abc\right]dududw\][/tex]

    now this looks like i have a domain that is a unit sphere and to make things easier i transform from the cartesian co-ordinate system to the spherical co-ordinate system with the standard transofrmation rules and i get,

    [tex]\[
    \intop_{0}^{2\pi}\intop_{0}^{\pi}\intop_{0}^{1}f\left(\frac{a}{r}cos\varphi sin\theta,\frac{b}{r}sin\varphi sin\theta,\frac{c}{r}cos\theta\right)\left[\left[abc\right]r^{2}sin\theta\right]drd\theta d\varphi\]
    [/tex]

    hence to confirm whether its right i just have to assume the function = 1 and if i integrate i must get the volume of the ellipsoid,

    [tex]\[
    \intop_{0}^{2\pi}\intop_{0}^{\pi}\intop_{0}^{1}\left[\left[abc\right]r^{2}sin\theta\right]drd\theta d\varphi\]
    [/tex]

    which is simple to integrate and which exactly gives me the volume of an ellipsoid [tex]=\frac{4}{3}\pi abc[/tex]

    2.Question - B

    Now my function is a dirac delta function that in itself which is a function of certain vectors. When I Integrate my dirac delta function over the ellipsoid as above i get strange results.
    thanx a lot.

    Tim
     
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
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