Definition:Differentiable Mapping

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Definition

Real Function

Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi$ be a point in $\left({a \,.\,.\, b}\right)$.


Definition 1

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

$\displaystyle \lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$

exists.


Definition 2

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

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

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 $r : U-x \to \R$ such that:

$(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + \alpha_1h_1 + \cdots + \alpha_nh_n + r\left({h}\right)\cdot h$
$(2):\quad$ $\displaystyle\lim_{h\to0} r(h) = 0$.


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 $r : U-x \to \R$ such that:

$(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + T(h) + r\left({h}\right)\cdot h$
$(2):\quad$ $\displaystyle\lim_{h\to0} r(h) = 0$.



Vector-Valued Function

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

Let $f = \left({f_1, f_2, \ldots, f_m}\right)^\intercal: \mathbb X \to \R^m$ 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:

$(1):\quad$ $\displaystyle f \left({x + h}\right) = f \left({x}\right) + T(h) + r\left({h}\right)\cdot \Vert h\Vert$
$(2):\quad$ $\displaystyle\lim_{h\to 0} r(h) = 0$.


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 $N$:

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

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



Also see