Definition:Differentiable Mapping between Manifolds

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Definition

Let $M$ and $N$ be differentiable manifolds.

Let $f : M \to N$ be continuous.

Definition 1

$f$ is differentiable if and only if for every pair of charts $(U, \phi)$ and $(V,\psi)$ of $M$ and $N$:

$\psi\circ f\circ \phi^{-1} : \phi ( U \cap f^{-1}(V)) \to \psi(V)$

Definition 2

$f$ is differentiable if and only if $f$ is differentiable at every point of $M$.

At a Point

Let $M$ and $N$ be differentiable manifolds.

Let $f: M \to N$ be continuous.

Let $p \in M$.

Definition 1

$f$ is differentiable at $p$ if and only if for every pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$:

$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$

is differentiable at $\phi \left({p}\right)$.

Definition 2

$f$ is differentiable at $p$ if and only if there exists a pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$ such that:

$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$

is differentiable at $\phi \left({p}\right)$.