# if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.

Prove that if $$f: X \rightarrow Y$$ is a diffeomorphism of manifolds with boundary, then $$\partial f$$ maps $$\partial X$$ diffeomorphically onto $$\partial Y$$.

Could anyone give me a hint for the proof of this please?

If you look at page 57 you'll find the definition of $$\partial X$$. Then, as usual, get the local diagram for $$f$$ - here $$U$$ is an open set in $$H^k$$ (p.14).
$$\begin{array}{lcl} X & \overset{f}{\longrightarrow} & Y \\ \downarrow\varphi & & \downarrow\psi \\ U & \overset{id}{\longrightarrow} & U \end{array}$$
Given commutativity of the diagram and the definition of $$\partial X$$, you should be able to prove it from here.
• what is $\partial f$ in this case? – Idonotknow Nov 26 '18 at 22:06
• $\partial f$ is $f$ restricted to $\partial X$, which is the image of boundary points in $U$ under $\phi^{-1}$. – Prototank Nov 26 '18 at 23:24