Vector-valued function

In mathematics, a vector-valued function is a function whose target set is a multidimensional vector space . Vector-valued functions are examined in particular in multi-dimensional analysis , differential geometry and functional analysis.

definition

One function

${\displaystyle f\colon D\to V}$

is called vector-valued if its target set is a vector space . In particular, the structure of the definition set is not relevant, only that of the target set. ${\displaystyle V}$ ${\displaystyle D}$

In many cases, the is used as the vector space ; such functions are then also called real-vector-valued . If the vector space is , then the functions are called analog complex-vector valued . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Examples

• The figure defined by ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ^{2}}$

${\displaystyle f(x)={\begin{pmatrix}x^{2}\\-3x\end{pmatrix}}}$

is a real vector valued function.

• The parametric representation of a curve in two or more dimensions is a real vector-valued function from to . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

• A vector-valued function is also called a vector field in this case . ${\displaystyle f\colon D\to \mathbb {R} ^{n}}$ ${\displaystyle D\subseteq \mathbb {R} ^{n}}$

literature

Otto Forster: Analysis 2 . Differential calculus im , ordinary differential equations. 10th, improved edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-02356-0 , doi : 10.1007 / 978-3-658-02357-7 . ${\displaystyle \mathbb {R} ^{n}}$