Solved

triple integral in spherical coordinates

Posted on 2006-10-28
2
1,114 Views
Last Modified: 2008-02-01
Consider the following triple integral:

     / 2pi   / pi       / sqrt(2)
    |        |           |               f(p,x,y) p^2 sin y dp dx dy
    |        |           |
   / 0      / 3pi/4   /0

Where p is rho, x is phi, y is theta in spherical coordinates. I'm trying to convert this integral to cartesian coordinates.

Its rather easy to see that the outer most integral will have limits of (-sqrt(2) to sqrt(2)) with respect to x, the middle integral will have limits of (-sqrt(2 - x^2) to sqrt(2 - x^2)) with respect to y, my problem is the inner most integral with respect to z. The book has the answer as something like -sqrt(2 - x^2 - y^2) to sqrt(x^2 + y^2). I'm just not seeing where this is coming from.


Thanks.
Brian
0
Comment
Question by:BrianGEFF719
2 Comments
 
LVL 84

Expert Comment

by:ozo
ID: 17827989
/ pi
 |
 |
 / 3pi/4
is a cone shaped spherical sector
so I would have expected the limits to be from sqrt(x^2 + y^2) to sqrt(2 - x^2 - y^2)
0
 
LVL 12

Accepted Solution

by:
jkmyoung earned 500 total points
ID: 17869185
since phi > pi/2 z is negative.
Since the radius is at most sqrt 2, this means that x² + y² + z² <= 2
z² <= 2 - x² - y²
-(2 - x² - y²) <= z <= 2 - x² - y²

Now for the restriction based on phi.

Note the conversion from &#952;, &#966;, r
x = r cos &#952; sin &#966;
y = r sin &#952; sin &#966;
z = r cos &#966;

cos (3pi/4) = -sqrt(2)        
sin (3pi/4) = sqrt(2)
cos (pi) = -1
sin (pi) = 0

so z must be at most -sqrt(x² + y²)
z <= -sqrt(x² + y²)
and since
x² + y² <= 2
<=> (2 - x² - y²) >= 0
=> -sqrt(x² + y²) <= (2 - x² - y²)

-(2 - x² - y²) <= z <= - (x² + y²)
I'm assuming you left the negative sign out of your orignal question.
0

Featured Post

Free Tool: SSL Checker

Scans your site and returns information about your SSL implementation and certificate. Helpful for debugging and validating your SSL configuration.

One of a set of tools we are providing to everyone as a way of saying thank you for being a part of the community.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
Inequality 4 84
Interest rate formula 7 70
Access question - internal training 9 70
Normalised finishing position 2 48
A Guide to the PMT, FV, IPMT and PPMT Functions In MS Excel we have the PMT, FV, IPMT and PPMT functions, which do a fantastic job for interest rate calculations.  But what if you don't have Excel ? This article is for programmers looking to re…
Have you ever thought of installing a power system that generates solar electricity to power your house? Some may say yes, while others may tell me no. But have you noticed that people around you are now considering installing such systems in their …
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201…

860 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question