(adsbygoogle = window.adsbygoogle || []).push({}); q1. The problem statement, all variables and given/known data

Let f : X ->Y, g : Y->Z be smooth. Show the composite is smooth. If f, g are

diffeomorphisms, so is the composite.

q2.Let B= {x : |x|^2 < a^2}. Show that

x -> ax/[(a^2 − |x|^2)^1/2]

is a diffeomorphism.

2. Relevant equations

3. The attempt at a solution

For q1 :A map is smooth if smooth functions pull back to smooth functions. If h : Z->R is

smooth, then by g’s smoothness ,so is hg, then by f’s smoothness so is hgf = h(gf).

Since this holds for all h, gf is smooth.

Actually i don't understand the answer and why one needs to come up with the function h.

For q2 : |f(x)| = a|x|/[(a^2 − |x|^2)^1/2]

and then rearrange symbols so that only |x| is on the right hand side

I don't know why I should start with the absolute value of f first ?

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# Homework Help: About diffeomorphism

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