# 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 $\struct {U, \phi}$ and $\struct {V, \psi}$ of $M$ and $N$ with $p \in U$ and $\map f p \in V$:

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

is differentiable at $\map \phi p$.

#### Definition 2

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

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

is differentiable at $\map \phi p$.