Directional derivative

In mathematics , the directional derivative of a function that is dependent on several variables is the instantaneous rate of change of this function in a direction given by a vector.

The Gâteaux differential is a generalization of the directional derivation to infinitely dimensional spaces .

Definitions

Be an open set , and a vector. ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle x\in {U}}$ ${\displaystyle v\in \mathbb {R} ^{n}}$

The directional derivative of a function at the point in the direction of is defined by the limit ${\displaystyle f\colon {U}\to \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle v}$

${\displaystyle D_{v}{f(x)}=\lim _{h\rightarrow 0}{\frac {f(x+h\cdot v)-f(x)}{h}},}$

if this exists.

Alternative definition

A function in the vicinity of 0 is defined by. is chosen so that the following applies ${\displaystyle f_{x,v}:t\mapsto f(x+t\cdot v)}$ ${\displaystyle f_{x,v}\colon (-\varepsilon ;\varepsilon )\to \mathbb {R} }$ ${\displaystyle \varepsilon >0}$

${\displaystyle \{x+t\cdot v|-\varepsilon <t<\varepsilon \}\subset U}$.

It is and the derivative of at the point is just the directional derivative of the point in the direction : ${\displaystyle f_{x,v}(0)=f(x)}$ ${\displaystyle f_{x,v}}$ ${\displaystyle t=0}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle v}$

${\displaystyle D_{v}f(x)=f_{x,v}'(0)={\frac {\mathrm {d} }{\mathrm {d} t}}f(x+t\cdot v){\Big |}_{t=0}}$

Unilateral directional derivatives

The unilateral directional derivatives of towards are defined by ${\displaystyle f\colon {U}\rightarrow \mathbb {R} }$ ${\displaystyle v\in \mathbb {R} ^{n}}$

${\displaystyle D_{v}^{+}{f(x)}=\lim _{h\rightarrow 0,\,h>0}{\frac {f(x+h\cdot v)-f(x)}{h}}}$

${\displaystyle D_{v}^{-}{f(x)}=\lim _{h\rightarrow 0,\,h<0}{\frac {f(x+h\cdot v)-f(x)}{h}}=-D_{-v}^{+}f(x).}$

The directional derivative in direction exists exactly when the two unilateral directional derivatives and coincide. In this case ${\displaystyle v}$ ${\displaystyle D_{v}^{+}{f(x)}}$ ${\displaystyle D_{v}^{-}{f(x)}}$

${\displaystyle D_{v}{f(x)}=D_{v}^{+}{f(x)}=D_{v}^{-}{f(x)}.}$

Derivation in normalized directions

Some authors define the directional derivative only in the direction of normalized vectors:

${\displaystyle D_{v}{f(x)}=\lim _{h\rightarrow 0}{\frac {f(x+h\cdot v)-f(x)}{h\cdot |v|}}.}$

These two definitions are the same for directions on the unit sphere . Otherwise the two definitions differ in terms of the factor . While the above definition is defined for all directions, the derivation in normalized directions is only defined for. ${\displaystyle v}$ ${\displaystyle \mathbb {S} ^{n-1}}$ ${\displaystyle |v|}$ ${\displaystyle v\neq 0}$

In applications in particular, it can be useful to calculate with the standardized direction vector ; this ensures that the directional derivative only depends on the direction, but not on the amount of . ${\displaystyle {\frac {v}{|v|}}}$ ${\displaystyle v}$

Spellings

Instead there are also the spellings ${\displaystyle D_{v}f(x)}$

${\displaystyle \nabla _{v}{f}(x)}$,   ,      And ${\displaystyle \partial _{v}f(x)}$ ${\displaystyle \displaystyle {\frac {\partial {f(x)}}{\partial {v}}}}$ ${\displaystyle f'_{v}(x)}$

common to avoid confusion with the covariant derivatives of differential geometry , among other things .

If totally differentiable , the directional derivative can be represented with the help of the total derivative (see the section Properties ). Spellings for it are ${\displaystyle f}$

${\displaystyle Df(x)v}$,   ,   ,      And . ${\displaystyle Df_{x}\,v}$ ${\displaystyle \operatorname {grad} \ f(x)\cdot v}$ ${\displaystyle \nabla f(x)\cdot v}$ ${\displaystyle (v\cdot \nabla )f(x)}$

properties

• If one chooses the coordinate unit vectors as the direction vector , one obtains the partial derivatives of in the respective point : ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}$ ${\displaystyle f}$ ${\displaystyle {\vec {x}}}$

  ${\displaystyle D_{{\vec {e}}_{i}}f({\vec {x}})={\frac {\partial f}{\partial x_{i}}}({\vec {x}})}$

• The one-sided directional derivative is homogeneous as a function of positive, i.e. for all positive the following applies: ${\displaystyle {\vec {v}}}$ ${\displaystyle \alpha >0}$

  ${\displaystyle D_{\alpha {\vec {v}}}^{+}{f({\vec {x}})}=\alpha D_{\vec {v}}^{+}{f({\vec {x}})}.}$

• If in is totally differentiable , the directional derivative as a function of is even linear and can be expressed by the gradient of : ${\displaystyle f}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle \nabla f}$ ${\displaystyle f}$

  ${\displaystyle D_{\vec {v}}{f}({\vec {x}})=\nabla f({\vec {x}})\cdot {\vec {v}}={\frac {\partial f}{\partial x_{1}}}({\vec {x}})\,v_{1}+\dots +{\frac {\partial f}{\partial x_{n}}}({\vec {x}})\,v_{n}}$

Examples

One-dimensional amount function

Absolute amount = its directional derivative in 0

In the one-dimensional case there are only two possible directions, namely to the left and to the right. The directional derivatives thus correspond to the usual one-sided derivatives. The derivatives in both directions can assume different values, which clearly means that the function can have a kink. A simple example of this is the amount function . It is not differentiable in, but the one-sided directional derivation exists: ${\displaystyle 0}$

${\displaystyle D_{v}^{+}{f(0)}=v\quad }$ For ${\displaystyle v\geq 0}$

and

${\displaystyle D_{v}^{+}{f(0)}=-v\quad }$ For ${\displaystyle v\leq 0}$

The absolute amount is therefore equal to its one-sided directional derivative in 0 as a function of . ${\displaystyle v}$

Normal derivation in areas

Is a smoothly bounded area with an outer normal vector field and , then is ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle (n>1)}$ ${\displaystyle \nu }$ ${\displaystyle f\in C^{1}({\bar {\Omega }})}$

${\displaystyle {\frac {\partial f}{\partial \nu }}=\nabla f\cdot \nu }$

the normal derivative of on the edge of . Objects of this type occur, for example, in partial differential equations with Neumann boundary conditions . ${\displaystyle f}$ ${\displaystyle \Omega }$

literature

• Otto Forster : Analysis 2. Differential calculus in R ⁿ . Ordinary differential equations. 7th edition. Vieweg-Verlag, 2006, ISBN 3-528-47231-6
• Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg 2000, ISBN 3-540-43580-8

Individual evidence

1. ^ Lothar Papula: Mathematics for engineers and natural scientists , Springer Verlag 2008, ISBN 978-3-8348-0225-5 , p. 66.