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August 2nd, 2017, 03:34 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3  changing of variables and determining the boundaries
Evaluate $\displaystyle \int \int (x+y) e^{xy} $ over the triangle with corners (0, 0), (1, 1), and (1, 1), using x = (u + v)/2, y = (u  v)/2 Is there any systematic method to determine the new boundaries of u and v? Last edited by zollen; August 2nd, 2017 at 04:30 PM. 
August 2nd, 2017, 04:26 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,404 Thanks: 1306 
you lost me at $x = (u+v)=2, ~y=(uv)=2$ do you mean $x = \dfrac{u+v}{2},~y = \dfrac{uv}{2}$ ? 
August 2nd, 2017, 04:30 PM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
Sorry, I just fixed my first post.

August 2nd, 2017, 05:20 PM  #4 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
You may have forgotten the Jacobian. No matter. You have already answered my question.

August 2nd, 2017, 05:58 PM  #5  
Senior Member Joined: Sep 2015 From: USA Posts: 2,404 Thanks: 1306  Quote:
But as you noticed they were off by a factor of 2 which is the determinant of the Jacobian, which in this case is just the transform matrix.  
August 2nd, 2017, 08:46 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,475 Thanks: 2039 
The triangle's vertices will be given by (u, v) = (0, 0), (0, 2) and (2, 0). The uaxis and the vaxis are perpendicular. Make a rough sketch.


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boundaries, changing, determining, variables 
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