Proving Smoothness of Composite Functions | Diffeomorphism q1 & q2

[Mr.M]

q1. Homework Statement
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.

Homework Equations

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 ?

 

[mighty2000]

Hello,

Thank you for your post. it is important to be able to clearly communicate your ideas and solutions to others. In your response, it would be helpful to provide more explanation and reasoning behind your answer. This will help others understand your thought process and learn from your approach.

For q1, the answer provided is using the definition of smoothness, which states that a map is smooth if smooth functions pull back to smooth functions. In other words, if a map takes a smooth function and produces another smooth function, then the map itself is smooth. The function h is used as an example of a smooth function, and by showing that hg and h(gf) are both smooth, it proves that gf is smooth.

For q2, the absolute value of f is used because the problem is asking to show that the given function is a diffeomorphism. A diffeomorphism is a smooth map with a smooth inverse. The absolute value is used to ensure that the function is well-defined for all values of x, including negative values. By showing that the inverse function exists and is also smooth, it proves that the given function is a diffeomorphism.

I hope this helps clarify the answers for you. Keep up the good work in your studies!