Strange change of variables

In summary, the author states that the change of variables for a multi-integral is not how it works and that one needs to use the jacobian method to express the change of variables.
  • #1

I'm reading an article where an integral of the form:

[tex] \int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\infty}_{-\infty}\,\mathrm{d}\tau''...[/tex]

The author then splits this into the region whereby [tex]\tau' >\tau'' [/tex], and the region [tex] \tau''>\tau' [/tex].

[tex] \int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\infty}_{\tau'}\,\mathrm{d}\tau''...+\int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\tau'}_{-\infty}\,\mathrm{d}\tau''...[/tex]

For the first integral the change of variables [tex] u=\tau'',s=\tau''-\tau' [/tex] is made, (and for the second part the change [tex] u=\tau',s=\tau'-\tau'' [/tex])

Focusing on just say the first integral for clarity, is this change of variables really well defined? I mean we could write [tex] \tau' =u-s, \tau''=u [/tex] and then we arrive at [tex] \mathrm{d}\tau'=du-ds, \mathrm{d}\tau''=du [/tex]. How does one uniquely change the s integral measure? My instincts say I should just use [tex] \mathrm{d}\tau'' \rightarrow du, \mathrm{d}\tau'\rightarrow -ds [/tex] but I don't really know how to justify this.

Secondly the limits, starting with the [tex] \tau'' [/tex] integral, [tex] \tau'' \in [\tau', +\infty]\rightarrow u \in [u-s ,+\infty] [/tex]...but the variable u can't be in it's own lower limit??

I can see by eye, if you like, that the max and min values of [tex] \tau' [/tex] are going to be [tex] u \in [-\infty,+\infty], s \in [0,\infty] [/tex], and indeed this is what the author has written, but how to show this analytically?

I tried to use Maple, just to see if it could do it using the change of variables commands, but it said it was "unable to solve the change of variable equations"
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  • #2
LAHLH said:
I mean we could write [tex] \tau' =u-s, \tau''=u [/tex] and then we arrive at [tex] \mathrm{d}\tau'=du-ds, \mathrm{d}\tau''=du [/tex]. How does one uniquely change the s integral measure? My instincts say I should just use [tex] \mathrm{d}\tau'' \rightarrow du, \mathrm{d}\tau'\rightarrow -ds [/tex] but I don't really know how to justify this.

When changing variables in a multi-integral, that's not how it works. In the double integral case you sort of have to think of it as an area integral dA, which in the current coordinates [itex]dA = d\tau'd\tau''[/itex]. When you change variables, though, dA needs to be expressed in terms of the new variables, [itex]dA = |\mathcal J| du ds[/itex], where [itex]|\mathcal J(u,s)|[/itex] is something called the Jacobian (or jacobian determinant, to be technically correct), which describes the scaling factor that accounts for the change of variables. You compute the jacobian as:

[tex]|\mathcal J(s,u)| = \left|\begin{array}{c c} \frac{\partial \tau'}{\partial u} & \frac{\tau''}{\partial u} \\ \frac{\partial \tau'}{\partial s} & \frac{\partial \tau''}{\partial s} \end{array}\right|[/tex]

Of course, since your changes of variables are rather simple (one of the variables you're just relabeling), you can get away with doing the one dimensional sort of thing: in the first term, the first integral is

[tex]\int_{\tau'}^\infty d\tau''[/itex]

As far as this integral is concerned, you can treat [itex]\tau'[/itex] as a constant:

[tex]\int_{\tau'-\tau'}^\infty d(\tau''-\tau') = \int_{0}^\infty ds[/itex]

But if the change of variables isn't that simple, you need to use the jacobian method.
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  • #3
Thanks a lot for the reply.

What is a strange change of variables?

A strange change of variables is a mathematical concept where the independent and dependent variables in a function are replaced with new variables, often in order to simplify the expression or make it easier to solve.

Why is it called "strange"?

It is called "strange" because the new variables used in the change of variables may seem unconventional or unexpected. This can sometimes lead to unexpected or counterintuitive results, making it a unique and interesting concept in mathematics.

When is a strange change of variables useful?

A strange change of variables can be useful in a variety of situations, including solving differential equations, transforming integrals, and simplifying complicated expressions. It can also be used in optimization problems and in finding patterns in data.

What are some examples of strange changes of variables?

One example is the substitution of variables in trigonometric functions, such as replacing sine and cosine with new variables such as u and v. Another example is the transformation of integrals using a new variable, such as x = 1/t, which can simplify the integral and make it easier to solve.

Are there any limitations to using a strange change of variables?

While a strange change of variables can be a powerful tool in mathematics, it is not always applicable or useful. The new variables used must still be consistent with the original problem and must not introduce any errors or inconsistencies. Additionally, the complexity of the problem may increase with the introduction of new variables.

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