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- Mar 10, 2012

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I have been reading some literature on manifolds, I recently changed the text book I was reading from and because of that things have gotten pretty messy. I have tried my best to extract the definitions from the text I am reading. Please tell me which ones I have got right and which ones I have not.

Thanks.

*Definition 1.*Let $f:\mathbb R^n\to \mathbb R^m$ be a function, $x$ be a point in $\mathbb R^n$ and $r>0$ be an integer. Then $f$ is said to be of

**class**$C^r$ at $x$ if all the partial derivatives of $f$ of order $r$ exist and are continuous at $x$. If $f$ if of class $C^r$ at each point in a subset $A\subseteq \mathbb R^n$ then we say that $f$ is of class $C^r$ in $A$. A continuous function is also said to be a $C^0$-

**function**.

**Definition 2.**Let $U$ and $V$ be open sets in $\mathbb R^n$. A function $h:U\to V$ is said to be a

**diffeomorphism**, or a $C^0$-

**diffeomorphism**, if $h$ is differentiable and has a differentiable inverse. For $r>0$, a diffeomorphism $h:U\to V$ is said to be of class $C^r$ if both $h$ and $h^{-1}$ are of class $C^r$.

**Definition 3.**A subset $M$ of $\mathbb R^n$ is said to be a $k$-

**dimensional manifold**of class $C^r$ in $\mathbb R^n$ if for all $x\in M$ there exist open sets $U$ and $V$ in $\mathbb R^n$ and a diffeomorphism $h:U\to V$ of class $C^r$ such that $x\in U$ and $$h(U\cap M)=V\cap (\mathbb R^k\times\{0\})$$