Is the Transition Map Smooth in the Intersecting Set?

sk1001
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Smooth transition map (easy!?)

Homework Statement


Check the transition map
http://img132.imageshack.us/img132/4341/18142532.png
is smooth in the set for which their images intersect

The Attempt at a Solution


I have thought of two ways to show this.

(1) Show that Φ is a composition of two smooth functions and is therefore smooth.
(2) compute the composite function and then prove that is smooth

Which way do you suggest?
I have attempted method (1) to some extent, but wondering if method (2) is easier.
 
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bump please
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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