Abstract
Consider a function \(f:H\rightarrow \mathbb {R}^{q}\), where H is an arbitrary set, and let the coordinates of the vector f(x) be denoted by \(f_1 (x),\dots , f_q (x)\) for every \(x\in H\). In this way we define the functions \(f_1 ,\dots , f_q\), where \(f_i :H\rightarrow \mathbb {R}\) for every \(i=1,\dots , q\). We call \(f_i\) the ith coordinate function or component of f .
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
Carl Jacobi (1804–1851), German mathematician.
- 2.
However, the inconsistencies would not have disappeared entirely. For \(p=1\) (i.e., for curves mapping to \(\mathbb {R}^{q}\)) the Jacobian matrix is a \(1\times q\) matrix, in other words, it is a column vector, while the derivative of the curve is a row vector.
- 3.
Leonhard Euler (1707–1783), Swiss mathematician.
- 4.
Jean Gaston Darboux (1842–1917), French mathematician. Darboux’s theorem states that if \(f:[a, b] \rightarrow \mathbb {R}\) is differentiable, then \(f'\) takes on every value between \(f'(a)\) and \(f'(b)\).
- 5.
The Bolzano–Darboux’s theorem states that if \(f:[a, b] \rightarrow \mathbb {R}\) is continuous, then f takes on every value between f(a) and f(b).
- 6.
Regarding the solution of equations using iterates, see Exercises 6.4 and 6.5 of [7]. In (a)–(d) of Exercise 6.4 the equations \(x=\sqrt{a+x},\ x=1/(2-x),\ x=1/(4-x),\ x=1/(1+x)\) are solved using iterates by defining sequences converging to the respective solutions. The solution of the equation \(x^2 =a\) using the same method can be found in Exercise 6.5.
- 7.
Stefan Banach (1892–1945), Polish mathematician.
- 8.
Joseph-Louis Lagrange (1736–1813), Italian-French mathematician.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media LLC
About this chapter
Cite this chapter
Laczkovich, M., Sós, V.T. (2017). Functions from \(\mathbb {R}^{p}\) to \(\mathbb {R}^{q}\) . In: Real Analysis. Undergraduate Texts in Mathematics, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7369-9_2
Download citation
DOI: https://doi.org/10.1007/978-1-4939-7369-9_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-7367-5
Online ISBN: 978-1-4939-7369-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)