Coordinate function

As coordinate functional be in linear algebra and in the topology designated special functions which the th component of a tuple , for example, the components of a supply column vector or the function value of a figure . ${\displaystyle i}$

definition

Be a -Tuple and . ${\displaystyle a\in \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle (a_{1},a_{2},\dotsc ,a_{n})}$ ${\displaystyle i\in \{1,\dotsc ,n\}}$

Then the -th coordinate function is defined as ${\displaystyle i}$ ${\displaystyle f_{i}\colon \mathbb {R} ^{n}\to \mathbb {R} }$

${\displaystyle f_{i}(a)=a_{i}}$.

The definition set and target set for can be defined differently depending on the context. ${\displaystyle f_{i}}$

topology

Be a map on a manifold with dimension . ${\displaystyle \varphi \colon U^{\varphi }\to A^{\varphi }}$ ${\displaystyle m}$

For a point there is then a -dimensional coordinate tuple in : ${\displaystyle p\in U^{\varphi }}$ ${\displaystyle \varphi (p)}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{m}}$

${\displaystyle \varphi (p)=(\varphi ^{1}(p),\varphi ^{2}(p),\dotsc ,\varphi ^{m}(p))}$.

So there are coordinate functions for a total of , each of which supplies the -th coordinate for . The superscript indices should not be confused with powers or the derivative. ${\displaystyle \varphi }$ ${\displaystyle m}$ ${\displaystyle \varphi ^{i},\;i\in \{1,\dotsc ,m\}}$ ${\displaystyle i}$ ${\displaystyle \varphi (p)}$

Individual evidence

1. ^ Frank Klinker: Fundamentals of Analysis. (PDF; 4.1 MB) p. 151 , accessed on July 5, 2019 . 
2. ^ Rolf Walter: Introduction to the differentiable manifolds. (PDF; 511 KB) July 15, 2009, p. 3 , accessed on July 5, 2019 .