# 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:

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

exists.

#### Definition 2

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

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

exists.

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

### Complex Function

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

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

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

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

$\ds \lim_{h \mathop \to 0} \frac {\map f {z_0+h} - \map f {z_0}} h$

exists as a finite number.

### Real-Valued Function

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

Let $\norm \cdot$ denote the Euclidean norm on $\R^n$.

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:

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

#### 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:

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

### 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:

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

#### 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 $x$.

### Function With Values in Normed Space

Let $U \subset \R$ be an open set.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.

A function $f : U \to X$ is (strongly) differentiable at $x \in U$ if and only if there exists $\map {f'} x \in X$ such that:

$\ds \lim_{h \mathop \to 0} \norm {\frac {\map f {x + h} - \map f x} h - \map {f'} x}_X = 0$

Moreover, $f$ is called (strongly) differentiable if it is differentiable at every point of $U$.

### 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 $\struct {U, \phi}$ 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 $\map \phi p$.