MHB Change of variables/ Transformations

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The discussion revolves around using variable transformations to manipulate equations of curves, specifically defining u = y/x and v = xy. The transformation leads to a defined square on the u-v plane with limits for integration set from 2 to 5 for the outer integral and 1 to 4 for the inner integral. A key point raised is the necessity of including the Jacobian in the new integral, which was initially overlooked. Participants emphasize the importance of verifying the Jacobian as a constant before proceeding with integration. The conversation concludes with a request for guidance on integrating the modified expression.
Joe20
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I have obtained as such using changing of variables/ transformations:

Let u= y/x and v= xy after manipulating the equation of the curves as such xy = 5, xy = 2, y/x = 1 and y/x = 4.
Then u = 1, u = 4, v = 5 and v = 2 => obtain a square on the u-v plane.

Change the integrand to:

Outer integral: 2 to 5
inner integral: 1 to 4
replace the expression for y/x to u [ u cos (u\pi) du dv]

I am not sure if I have done it correctly thus far. If so, how do i integrate that expression? Need advice.
 

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Alexis87 said:
I have obtained as such using changing of variables/ transformations:

Let u= y/x and v= xy after manipulating the equation of the curves as such xy = 5, xy = 2, y/x = 1 and y/x = 4.
Then u = 1, u = 4, v = 5 and v = 2 => obtain a square on the u-v plane.

Change the integrand to:

Outer integral: 2 to 5
inner integral: 1 to 4
replace the expression for y/x to u [ u cos (u\pi) du dv]

I am not sure if I have done it correctly thus far. If so, how do i integrate that expression? Need advice.

Seems to me you'll be needing the Jacobian in your new integral. I'm hoping this isn't the first you've heard of it.
 
tkhunny said:
Seems to me you'll be needing the Jacobian in your new integral. I'm hoping this isn't the first you've heard of it.
Hi Yes I will need to multiply by the jacobian value which I have missed out. (anyway that suppose to be a constant). Would like to know how should I integrate such expression from what I have stated.
 
Alexis87 said:
Hi Yes I will need to multiply by the jacobian value which I have missed out. (anyway that suppose to be a constant). Would like to know how should I integrate such expression from what I have stated.

No objection to anything else I see. You had better PROVE that it's a constant.
 

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