Solved

triple integral in spherical coordinates

Posted on 2006-10-28
2
1,111 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

ScreenConnect 6.0 Free Trial

At ScreenConnect, partner feedback doesn't fall on deaf ears. We collected partner suggestions off of their virtual wish list and transformed them into one game-changing release: ScreenConnect 6.0. Explore all of the extras and enhancements for yourself!

Question has a verified solution.

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

Suggested Solutions

Title # Comments Views Activity
Roulette strategy probability question 58 99
Forces on a string 9 59
Math Stumper 6 51
what causes a soft drink to evaporate inside a sealed aluminum can over time? 7 71
This article seeks to propel the full implementation of geothermal power plants in Mexico as a renewable energy source.
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tec…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
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…

770 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