Total differentiability

The total differentiability is in the mathematical branch of analysis a fundamental property of functions between finite-dimensional vector spaces over . By means of this property, many other statements about functions that are important for analysis can be shown. (These statements are not valid when using the weaker partial differentiability , which is formally more similar to the usual definition of the differentiability of a real function as the convergence of the difference quotients.) Many other terms of analysis are then based on total differentiability. In the more recent mathematical literature one usually speaks of differentiability instead of total differentiability. ${\displaystyle \mathbb {R} }$

The total differentiability of a function at a point means that it can be approximated locally by a linear mapping , while the partial differentiability (in all directions) only the local approximability by straight lines in all coordinate axis directions, but not as a single linear one Figure calls.

While the derivative of a function at one point is usually understood as a number, in the higher-dimensional case the derivative is understood as that local linear approximation. This linear mapping can be represented by a matrix called the derivative matrix , Jacobian matrix, or fundamental matrix (in the one-dimensional case this again results in a 1 × 1 matrix, i.e. a single number). In the one-dimensional case, the classical real, total and partial differentiability concepts agree. ${\displaystyle \,\!f'(x_{0})}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x_{0}\in \mathbb {R} }$

The concept of Fréchet differentiability generalizes the total differentiability to infinitely dimensional spaces, it adopts the property of derivation as a local, linear approximation.

Motivation / introduction

For functions , the derivation at the point is usually through ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x_{0}}$

${\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}=\lim _{h\to 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}}$

defined, with or . In this form, the definition cannot be transferred to figures , since one can not divide by. One therefore pursues a different path. ${\displaystyle h=x-x_{0}}$ ${\displaystyle x=x_{0}+h}$ ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle h\in \mathbb {R} ^{n}}$

The derivative describes the slope of the tangent to the function graph at the point . The tangent itself has the equation ${\displaystyle \,f'(x_{0})}$ ${\displaystyle (x_{0},f(x_{0}))}$

${\displaystyle \,y=f(x_{0})+f'(x_{0})(x-x_{0}),}$

it is therefore the graph of the linear (affine) function

${\displaystyle x\mapsto f(x_{0})+f'(x_{0})(x-x_{0})}$.

This function approximates the function in the following sense: ${\displaystyle f}$

${\displaystyle \,f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+r(x-x_{0})}$

or (with , also ) ${\displaystyle h=x-x_{0}}$ ${\displaystyle x=x_{0}+h}$

${\displaystyle \,f(x_{0}+h)=f(x_{0})+f'(x_{0})h+r(h)}$,

where the error term approaches 0 for faster than , that is ${\displaystyle r(h)}$ ${\displaystyle h\to 0}$ ${\displaystyle h}$

${\displaystyle \lim _{h\to 0}{\frac {|r(h)|}{|h|}}=0.}$

In this form, the concept of differentiability can be transferred to images . In this case there is a vector in , a vector in and a linear mapping from to . ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle h}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle F(x_{0}+h)-F(x_{0})}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle \,F'(x_{0})}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$

definition

Let an open subset , a point and a map be given . The mapping is called (totally) differentiable at the point , if a linear mapping ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ ${\displaystyle x_{0}\in U}$ ${\displaystyle F\colon U\to \mathbb {R} ^{m}}$ ${\displaystyle F}$ ${\displaystyle x_{0}}$

${\displaystyle L\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$

exists that the figure

${\displaystyle h\mapsto F(x_{0}+h)-F(x_{0})}$

approximated, that is, for the "error function"

${\displaystyle r(h)=F(x_{0}+h)-F(x_{0})-L(h)}$

applies

${\displaystyle \lim _{h\to 0}{\frac {\|r(h)\|}{\|h\|}}=0.}$

A vector in . The double dashes denote a vector norm in or . Since im or all standards are equivalent, it does not matter which standard is selected. ${\displaystyle h}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$

If such a linear mapping exists, it is uniquely determined. It is called the ( total) differential or simply the derivative of the point and writes it , , or . ${\displaystyle L}$ ${\displaystyle F}$ ${\displaystyle x_{0}}$ ${\displaystyle DF(x_{0})}$ ${\displaystyle DF_{x_{0}}}$ ${\displaystyle dF_{x_{0}}}$ ${\displaystyle F'(x_{0})\,}$

Conversely, if in a neighborhood of all partial derivatives of exist and in are continuous, the (total) differentiability of in already follows . ${\displaystyle x_{0}}$ ${\displaystyle F}$ ${\displaystyle x_{0}}$ ${\displaystyle F}$ ${\displaystyle x_{0}}$

See also

• Total differential , for the derivation concept for real-valued functions based on total differentiability.
• Pushforward , the derivation term that results from the total differentiability for functions between differentiable manifolds .

literature

• Konrad Königsberger : Analysis 2 . Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 .