Integration over an Ellipsoidal Domain - Clarification

In summary: Your Name]In summary, Tim is seeking assistance with integrating a function over an ellipsoidal domain. He has followed a correct approach using transformation of variables and has confirmed it by integrating with a function equal to 1. However, when dealing with a dirac delta function, he needs to use distributional derivatives and consult with a mathematics expert for further assistance.
  • #1
tim85ruhruniv
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Homework Statement



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.

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
 
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  • #2


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.


 

FAQ: Integration over an Ellipsoidal Domain - Clarification

What is integration over an ellipsoidal domain?

Integration over an ellipsoidal domain is a mathematical technique used to calculate the volume under a three-dimensional curved surface, known as an ellipsoid. It involves breaking down the ellipsoid into smaller, simpler shapes and summing their volumes to find the total volume of the ellipsoid.

What is the purpose of integration over an ellipsoidal domain?

The purpose of integration over an ellipsoidal domain is to accurately calculate the volume of complex three-dimensional shapes, such as ellipsoids. This technique is often used in fields such as physics, engineering, and geology to model and analyze objects with curved surfaces.

How is integration over an ellipsoidal domain performed?

Integration over an ellipsoidal domain is typically performed using calculus, specifically the technique of triple integration. This involves setting up three nested integrals, one for each dimension of the ellipsoid, and using appropriate limits of integration to cover the entire volume of the shape.

What are some applications of integration over an ellipsoidal domain?

Integration over an ellipsoidal domain has many practical applications. It is commonly used in geodesy to model the shape of the Earth, in astronomy to calculate the volume of planets and other celestial bodies, and in engineering to design and analyze complex structures with curved surfaces.

What are some challenges associated with integration over an ellipsoidal domain?

One of the main challenges of integration over an ellipsoidal domain is setting up the appropriate limits of integration for the nested integrals. This can be a complex process, especially for irregularly shaped ellipsoids. Additionally, numerical errors can arise due to the large number of calculations involved in this technique.

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