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