Integration over an Ellipsoidal Domain - Clarification

[tim85ruhruniv]

Homework Statement

I want to integrate a function over an Ellipsoidal domain.

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

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.

$$\[ x=ua,y=vb,z=wc\]$$

and hence i get,

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

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,

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

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,

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

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

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

 

[mighty2000]

Dear Tim,

Thank you for sharing your solution for integrating a function over an ellipsoidal domain. Your approach seems to be correct and it is great that you have checked it by integrating with a function that is equal to 1. However, when dealing with a dirac delta function, it is important to keep in mind that it is a distribution rather than a traditional function. This means that it cannot be evaluated at a single point and its integral over a domain is not well-defined.

In order to integrate a dirac delta function over an ellipsoidal domain, you will need to use the concept of distributional derivatives and generalize your integration approach. This can be a bit more complex and may require some advanced mathematical techniques. I would recommend consulting with a mathematics expert for assistance with this specific problem.

I hope this helps and good luck with your integration! Keep up the good work.