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 .${\ displaystyle \ mathbb {R} ^ {n}}$