Differentiable

Statements

Definition. (Citation Needed) Let $M$ and $N$ be two differentiable manifolds of dimension $m$ and $n$, respectively. A map $f:M\to N$ is said to be differentiable (or smooth) at a point $p\in M$ if there exists parametrizations $(U,\phi )$ of $M$ at $p$ (i.e. $p\in \phi (U)$) and $(V,\psi )$ of $N$ at $f(p)$ with $f(\phi (U))\subset \psi (V)$ such that the map

$$\hat{f}:={\psi }^{-1}\circ f\circ \phi :U\subset {\mathbb{R}}^m\to {\mathbb{R}}^n$$

is smooth.

Remarks

When $M$ and $N$ are Euclidean spaces, the previous definitions reduces to the following. Let $X\subset {\mathbb{R}}^n$ be an open set the vector-valued function $F:X\to {\mathbb{R}}^m$ is said to be differentiable at $a\in X$ if there exists an $m\times n$ matrix $A$ and an $m-$tuple $R(x,a)=(r^1(x,a),\dots ,r^m(x,a))$ defined on $X\times X$ such that the limit ${\lim }_{x\to a}||R(x,a)||=0$ holds and, for every $x\in X$, the equation

F(x)=F(a)+A(x-a)+||x-a||R(x,a) \end{equation} $$^differentiable also holds. When $m=1$, the $m-$tuple reduces to $r(x,a)$, the matrix $A$ reduces to the vector $b=(b_1,\ldots,b_n)$ and the Equation (differentiable) reduces to $$f(x)=f(a)+\sum_{i=1}^n b_i(x^i-a^i)+||x-a||r(x,a)\;.$$ ## References - [[L. Godinho, J. Natario - An introduction to riemannian geometry|@GodNat14, Definition 3.1]] - [[Boothby, W. M. - An introduction to differentiable manifolds and riemannian geometry|@Boo75 pp. 21 and Definition 2.1]]