Change of variables/ Transformations

In summary, the conversation discusses the process of obtaining a square on the u-v plane using changing of variables/transformations. The equation of the curves is manipulated to obtain values for u and v, and the integrand is changed accordingly. The use of the Jacobian is mentioned and the need to prove that it is a constant. The speaker also mentions needing advice on how to integrate the expression.
  • #1
Joe20
53
1
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

FAQ: Change of variables/ Transformations

What is a change of variables/transformation?

A change of variables or transformation is a mathematical technique used to convert a given set of variables into a different set of variables, while preserving the relationship between them. It is often used to simplify complex equations or to solve problems in different coordinate systems.

Why is change of variables/transformation important in science?

Change of variables/transformation is important in science because it allows us to change the perspective from which we view a problem, making it easier to analyze and solve. It is also commonly used in physics and engineering to simplify equations and make calculations more manageable.

What types of transformations are commonly used in science?

There are various types of transformations used in science, including linear transformations, logarithmic transformations, polar transformations, and Fourier transformations. Each type has its own specific purpose and can be used to solve different types of problems.

How do you determine the appropriate change of variables/transformation to use?

The appropriate change of variables/transformation to use depends on the specific problem you are trying to solve. It is important to analyze the given variables and their relationships to determine which type of transformation will best simplify the problem and lead to a solution.

Are there any limitations or drawbacks to using change of variables/transformation?

While change of variables/transformation can be a powerful tool in solving problems, it is not always applicable. In some cases, the transformation may not be possible or may lead to a more complex problem. It is important to carefully consider the limitations and drawbacks before using this technique in a scientific study.

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