Showing that $X^{-1}_{2}\circ {X_1}:X^{-1}_1(M_1\cap M_2)\to X^{-1}_2(M_1\cap M_2)$ is smooth for parametrizations of manifolds that intersect

Question

I have been spending a lot of time trying to show the function $$X^{-1}_{2}\circ {X_1}:X^{-1}_1(M_1\cap M_2)\to X^{-1}_2(M_1\cap M_2)$$ is $C^1$, where $X_i:U_i(\subset \Bbb{R}^k)\to M_i$ are parameterizations to two manifolds $M_i\subset \Bbb{R}^d$ and $M_1\cap M_2\ne \emptyset$. I am struggling with computing the partial derivative ${{\partial((X^{-1}_2\circ X_1)_i)(u)}\over {\partial u_j}}={{\partial((X^{-1}_2)_i(X_1(u))}\over {\partial u_j}}=\sum _{m=1}^{d}{{\partial (X^{-1}_2)_i(X_1(u))} \over {\partial (X_1)_m(u)}}\cdot {{\partial (X_1)_m(u)} \over {\partial u_j}}$ and it's becoming very confusing, and I can't show that any component is continuous. I thought that I should try to represent one of the multivariable functions as a function of one variable where all the others are fixed. Still, the complicated change of variables it requires makes me think that my approach is wrong. Can you help me get the partial derivative computation correct, or give me a clue on how to show smoothness without derivation?

Answer 1

This is an attempt to answer my question, so if there are any false arguments please let me know. In order to show $X_2^{-1}$ is smooth, I will show that ${\partial (X_2^{-1})_m(X_1(u))\over \partial u_j}$ is defined and continuous.

First, note that $X_2^{-1}$ is continuous. By definition, ${\partial (X_2^{-1})_m(X_1(u))\over \partial u_j}=\sum_{i=1}^{d}{\partial (X_2^{-1})_m(X_1(u))\over \partial (X_1)_i(u)}\cdot {\partial (X_1)_i(u)\over \partial u_j}$, so it comes down to showing that the expression ${\partial (X_2^{-1})_m(X_1(u))\over \partial (X_1)_i(u)}$ is continuous.

By setting $v=X_1(u)$, ${\partial (X_2^{-1})_m(X_1(u))\over \partial (X_1)_i(u)}={\partial (X_2^{-1})_m(v)\over \partial (v_i)}|_{X_1(u)}$.

Now set $X_2(w)=X_1(u)$, then ${\partial (X_2^{-1})_m(X_1(u))\over {\partial (X_1)_i(u)}}={\over{{\partial { (X_1)_i(u)}\over \partial (X_2^{-1})_m (X_2(u))}}}={1\over {{\partial (X_2)_i(w)}\over {\partial w_m}}}$ where the last result is well-defined since $X_2$ is supposed to be smooth and it is defined on $w$, so it is continuous there.