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I am currently in a graduate level real analysis course, and we have been working with the Inverse and Implicit Function Theorem. With these problems, I often find myself struggling to set the work up correctly (ie find the right function to apply these theorems to, etc). Are there any general techniques that you have found useful for these types of problems?

For example, we had the following problem: Let $S$ and $T$ be surfaces in $\mathbb{R}^n$ defined respectively by $s(x)=0$ and $t(x)=0$, where $s$ and $t$ are $C^1$ functions mapping $\mathbb{R}^n$ to $\mathbb{R}$. Suppose $s(a)=t(b)=0$ for some points $a,b$ in $\mathbb{R}^n$, and suppose $s'(a)\not=0$, $t'(b)\not=0$. Show that there are open neighborhoods $U$ and $B$ in $\mathbb{R}^n$ containing $a$ and $b$ respectively, and a $C^1$ diffeomorphism $\phi :U\to V$ such that $\phi(S\cap U)= T\cap V$ and $\phi (a)=b$.

For this problem, I could tell that I would want to use the Inverse Function theorem because we are looking for a diffeomorphism. However, applying the Inverse Function theorem directly to $s$ or $t$ would not work, as the domains of $s$ and $t$ are $\mathbb{R}^n$ and they map into $\mathbb{R}$. Thus I got stuck on trying to finding maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ that would give me anything useful. The teacher told us to use maps $F_s(x_1, x_2,...,x_n)=(x_1-a_1+b_1, x_2-a_2+b_2,..., x_{n-1}-a_{n-1}+b_{n-1}, s(x))$ and $F_t(x_1, x_2,...,x_n)=(x_1, x_2,..., x_{n-1}, t(x))$. After seeing this, I was able to easily solve the problem.

It seems to me that expanding the image of a function in this way might be a common technique in inverse function theorem problems, as we also used this technique in another problem. Thus I was wondering if there were any other weird little techniques like this that others might know of.

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  • $\begingroup$ Can you give an example problem, what you tried, and what difficulties you encountered? $\endgroup$ – Acccumulation Nov 7 '17 at 22:46
  • $\begingroup$ Edited to include an example $\endgroup$ – Samantha M Nov 7 '17 at 23:29

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