Diffeomorphisms from Open Sets in Rm to Open Sets in Rn

Diffeomorphisms from Open Sets in Rm to Open Sets in Rn

Definition: Let $U \subseteq \mathbb{R}^m$ and $V \subseteq \mathbb{R}^n$. A function $F : U \to V$ is called a Diffeomorphism from $U$ to $V$ if $F$ has the following properties:
a) $F : U \to V$ is bijective.
b) $F : U \to V$ is smooth.
c) $F^{-1} : V \to U$ is smooth.

A function $F : U \to V$ is said to be smooth if $F$ is infinitely differentiable and then set of all smooth functions from $U$ to $V$ is denoted $C^{\infty} (U, V)$.

Let's look at a very simple example of a diffeomorphism. Let $U = \mathbb{R}$ and $V = (0, \infty)$. Define a function $F : U \to V$ for all $x \in U$ by:

(1)
\begin{align} \quad F(x) = e^x \end{align}

Then $F$ is clearly a bijection. Furthermore, $F$ is smooth since $e^x$ is infinitely differentiable as we known from elementary calculus and for all $n \in \mathbb{N}$, $F^{(n)}(x) = e^x$.

Moreover we have that $F^{-1} : V \to U$ defined by $F^{-1}(x) = \ln x$ is also smooth, since $(F^{-1})^{(1)}(x) = x^{-1}$, $(F^{-1})^{(2)}(x) = -x^{-2}$, $(F^{-1})^{(3)}(x) = 2x^{-3}$, and in general:

(2)
\begin{align} \quad (F^{-1})^{(n)} = (-1)^{n-1}(n - 1)!x^{-n} \end{align}

So indeed $F$ is a diffeomorphism from $U$ to $V$. The graphs of $F$ (red) and $F^{-1}$ (blue) are given below.

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