Change of variables/ Transformations

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Discussion Overview

The discussion revolves around the process of changing variables or transformations in the context of integration, specifically using the variables u and v derived from the original variables x and y. Participants explore the implications of these transformations on the integrals and the necessity of including the Jacobian in the calculations.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant describes their approach to changing variables, defining u = y/x and v = xy, and outlines the resulting bounds for integration.
  • Another participant emphasizes the need to include the Jacobian in the new integral, suggesting it is a crucial step that should not be overlooked.
  • A participant acknowledges the omission of the Jacobian and expresses a desire for guidance on how to integrate the expression they have formulated.
  • There is a challenge posed regarding the assertion that the Jacobian is a constant, indicating a need for proof of this claim.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of using the Jacobian in the transformation process, but there is no consensus on whether the Jacobian can be considered a constant without further justification.

Contextual Notes

Participants have not fully resolved the implications of the Jacobian in their integrals, and there are unresolved questions regarding the integration process itself.

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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