I A computation of an integral on page 344 of Schutz's textbook

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On page 344 of "A First Course in GR" he writes the following:
Consider, next, ##k=1##. Let us define a new coordinate ##\chi(r)## such that:
$$(12.16)\ \ \ d\chi^2 = \frac{dr^2}{1-r^2}$$
and ##\chi =0 ## where ##r=0##. This integrates to
$$(12.17) \ \ \ r=\sin \chi,$$
When I do the integration I get the following: ##\int_0^{\chi^2}d\chi^2= \int_0^{r^2}\frac{dr^2}{1-r^2}= \chi^2 = -\ln (1-r^2)##, after I invert the last relation I get: ##r=\sqrt{1-\exp(-\chi^2)}##, where did I go wrong in my calculation?

Thanks!
 
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You should of course integrate
$$\int \mathrm{d} \chi = \int \mathrm{d}r \frac{1}{\sqrt{1-r^2}}...$$
 
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vanhees71 said:
You should of course integrate
$$\int \mathrm{d} \chi = \int \mathrm{d}r \frac{1}{\sqrt{1-r^2}}...$$
Ah, yes you are correct. It's ##(dx)^2## and not ##d(x^2)##.
I am really getting old and sloppy... :-(
 
How do you define and Integral with ##\mathrm{d} x^2##, and it's not ##\mathrm{d} (x^2)=2x \mathrm{d} x##, because you want ##\chi## and not something else ;-).
 
vanhees71 said:
How do you define and Integral with ##\mathrm{d} x^2##, and it's not ##\mathrm{d} (x^2)=2x \mathrm{d} x##, because you want ##\chi## and not something else ;-).
I meant I thought it was ##d(\chi^2)=\frac{d(r^2)}{1-r^2}##. But now I see my mistake, thanks for clearing this simple thing.
 
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