Definition:Differentiable Mapping

Definition

Real Function

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi$ be a point in $\openint a b$.

Definition 1

$f$ is differentiable at the point $\xi$ if and only if the limit:

$\displaystyle \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$

exists.

Definition 2

$f$ is differentiable at the point $\xi$ if and only if the limit:

$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$

exists.

These limits, if they exist, are called the derivative of $f$ at $\xi$.

Complex Function

Let $D\subset \C$ be an open set.

Let $f : D \to \C$ be a complex function.

Let $z_0 \in D$ be a point in $D$.

Then $f$ is complex-differentiable at $z_0$ if and only if the limit:

$\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$

exists as a finite number.

Real-Valued Function

Let $U$ be an open subset of $\R^n$.

Let $f: U \to \R$ be a real-valued function.

Let $x \in U$.

Definition 1

$f$ is differentiable at $x$ if and only if there exist $\alpha_1, \ldots, \alpha_n \in \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that:

$(1):\quad \map f {x + h} = \map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n + \map r h\cdot h$
$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

Definition 2

$f$ is differentiable at $x$ if and only if there exists a linear transformation $T: \R^n \to \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that:

$(1):\quad \map f {x + h} = \map f x + \map T h + \map r h \cdot h$
$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

Vector-Valued Function

Let $\mathbb X$ be an open subset of $\R^n$.

Let $f = \tuple {f_1, f_2, \ldots, f_m}^\intercal: \mathbb X \to \R^m$ be a vector valued function.

Definition 1

$f$ is differentiable at $x \in \R^n$ if and only if there exists a linear transformation $T: \R^n \to \R^m$ and a mapping $r : U \to \R^m$ such that:

$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$
$(2): \quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$

Definition 2

$f$ is differentiable at $x \in \R^n$ if and only if for each real-valued function $f_j: j = 1, 2, \ldots, m$:

$f_j: \mathbb X \to \R$ is differentiable at $x$.

Between Differentiable Manifolds

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 $\phi \left({p}\right)$.