Change of variable for Jacobian: is there a method?

In summary, the conversation discusses choosing the correct change of variables to map a given region to a rectangular region. The correct choice is a), which uses a substitution for variable u and a trivial substitution for variable v. There is no systematic way to determine the appropriate change of variables, but one can observe that the new region is in the 1st Quadrant and that the chosen substitution shifts the rectangle 1 unit in the positive u direction. There is also a discussion about other common substitutions used in similar problems.
  • #1
fatpotato
Homework Statement
Choose the correct change of variables among the three possible choices to map ##\Sigma## to a rectangular region.
Relevant Equations
None for now
Hello,

This problem comes just prior to introducing change of variables with Jacobian.

Given the following region in the x-y plane, I have to choose (with justification) the correct change of variables associated, for ##u\in [0,2]## and ##v \in [0,1]##.

The correct choice here is a), but I do not understand why...Solution only mentions that a possible change of variable for x would be ##x = -(y+1) + u(y+1)##, so we get to solution a).

Here is my question: is there a systematic way of finding the appropriate change of variables? How did the instructor suddenly decide that choosing the previous substition would be a good idea to solve the problem? I have no idea where to begin.

Thank you very much for your help.
 

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  • #2
fatpotato said:
Homework Statement:: Choose the correct change of variables among the three possible choices to map ##\Sigma## to a rectangular region.
Relevant Equations:: None for now

Hello,

This problem comes just prior to introducing change of variables with Jacobian.

Given the following region in the x-y plane, I have to choose (with justification) the correct change of variables associated, for ##u\in [0,2]## and ##v \in [0,1]##.

The correct choice here is a), but I do not understand why...Solution only mentions that a possible change of variable for x would be ##x = -(y+1) + u(y+1)##, so we get to solution a).
You can at least verify that ##u=0## and ##u=2## correspond to the left-hand boundary and the right-hand boundary, respectively, for the region in the x-y plane.
Also, ##u=1## corresponds to the vertical line, ##x=0## which passed through the middle of the region.

Here is my question: is there a systematic way of finding the appropriate change of variables? How did the instructor suddenly decide that choosing the previous substitution would be a good idea to solve the problem? I have no idea where to begin.

Thank you very much for your help.
It's hard to guess how the instructor decided on that substitution (whether it was sudden or not). It's also difficult to guess the reason for choosing that particular region for the new variables, except to observe that the new region is in the 1st Quadrant.

Let's examine choice a), with emphasis on the defined mapping for new variable, ##u##. - The mapping for variable ##v## is trivial.

Rewrite
##\displaystyle u=\frac{ x+y+1 }{ y+1 } ##

##\quad \displaystyle=\frac{ x }{ y+1 } + \frac{ 1 }{ y+1 } ##

##\quad \displaystyle=\frac{ x }{ y+1 } +1 ##

The effect of that final term of 1 is to shift the rectangle 1 unit in the positive ##u## direction.

Let's drop that final ## + 1 ## and change the variable name from ##u## to ##n## so as to avoid confusion. We now have:
## \displaystyle n=\frac{ x }{ y+1 } ##.

Solving for ##x## gives : ## \displaystyle x=n\left(y+1 \right) ##.

If you think of the rather unusual practice of graphing ##x## as a function of ##y##, we have that this gives a line with slope ##n## and ##y## intercept of ##-1## .

Any such line with slope ## n \in [ -1,~1]## will intersect the trapezoid at all values of ##y \in [0,~1]## and nowhere else.
 
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  • #3
Hello,

Thank you for your answer. I did not think about formulating ##u## as a parameter ##n##! This gives a nice parametric linear equation and indeed gives insight on the area's border.

SammyS said:
It's hard to guess how the instructor decided on that substitution (whether it was sudden or not).
I guess that there are usual tricks to know and to try on typical problems...Going through a Schaum on change of variables with Jacobian matrices, I noticed the prevalence of certain substitutions, like ##x=u+v##, ##y=u-v## and the classical polar coordinates substitution.
 
  • #4
You might begin by noting that ##u## is the same in all three options so all that we must do is to choose the appropriate ##v##, since ##x## is bounded by functions of ##y## we know (b) is incorrect. The remaining two give ##v=\pm y## and being as it is given that ##v\in \left[ 0,1\right]## and ##0\le y\le 1## we must have ##v=y## (throwing out the negative) or change of variables (a).
 
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  • #5
benorin said:
You might begin by noting that ##u## is the same in all three options so all that we must do is to choose the appropriate ##v##, since ##x## is bounded by functions of ##y## we know (b) is incorrect. The remaining two give ##v=\pm y## and being as it is given that ##v\in \left[ 0,1\right]## and ##0\le y\le 1## we must have ##v=y## (throwing out the negative) or change of variables (a).
This is indeed clever, however there is a twist.

I cropped the solution picture: there should be an additional answer d) for "None of the above". I cropped the last choice because it was not in english. But I still agree with you, it is more efficient to only check for the substitution for ##v## in this case.
 
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  • #6
In that case you need only narrow it down to (a) or (d) and then draw out the transformed boundaries to see which of the two options is correct.
 
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1. What is a change of variable for Jacobian?

A change of variable for Jacobian is a mathematical technique used to transform an integral from one coordinate system to another. It involves substituting variables in an integral to make it easier to solve or evaluate.

2. Why is a change of variable for Jacobian useful?

A change of variable for Jacobian can simplify complex integrals and make them easier to solve. It also allows for integrals to be evaluated in different coordinate systems, providing a more versatile approach to solving mathematical problems.

3. How do you perform a change of variable for Jacobian?

To perform a change of variable for Jacobian, you need to first identify the original coordinate system and the desired coordinate system. Then, you need to find the appropriate transformation equations and apply them to the integral. Finally, you can solve the integral using the new variables.

4. Are there any limitations to using a change of variable for Jacobian?

Yes, there are limitations to using a change of variable for Jacobian. The transformation equations must be well-defined and invertible, and the Jacobian determinant must be non-zero in order for the method to work. Additionally, some integrals may not have closed-form solutions even after a change of variable is applied.

5. Can a change of variable for Jacobian be used for any type of integral?

No, a change of variable for Jacobian is typically used for integrals involving multiple variables. It may not be necessary or useful for single variable integrals. Additionally, the method may not be applicable to certain types of integrals, such as improper integrals or those with infinitely many variables.

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