Mathematical Analysis

An Introduction to Functions of Several Variables

verfasst von: Mariano Giaquinta, Giuseppe Modica

Verlag: Birkhäuser Boston

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This self-contained work is an introductory presentation of basic ideas, structures, and results of differential and integral calculus for functions of several variables.

The wide range of topics covered include: differential calculus of several variables, including differential calculus of Banach spaces, the relevant results of Lebesgue integration theory, differential forms on curves, a general introduction to holomorphic functions, including singularities and residues, surfaces and level sets, and systems and stability of ordinary differential equations. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.

Mathematical Analysis: An Introduction to Functions of Several Variables motivates the study of the analysis of several variables with examples, observations, exercises, and illustrations. It may be used in the classroom setting or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering.

Other books recently published by the authors include: Mathematical Analysis: Functions of One Variable, Mathematical Analysis: Approximation and Discrete Processes, and Mathematical Analysis: Linear and Metric Structures and Continuity, all of which provide the reader with a strong foundation in modern-day analysis.

Inhaltsverzeichnis

Frontmatter

1. Differential Calculus

In this chapter we discuss the basic notions of differential calculus of functions of several variables.

2. Integral Calculus

The problems of characterizing the class of functions that are Riemann integrable and of discussing discontinuous functions, in particular of understanding for which functions the fundamental theorem of calculus is valid, as well as the need of integrating new functions, led to a new definition of integral due to Henri Lebesgue (1875–1941). Though the main ideas of Lebesgue's integration theory go back to Henri Lebesgue (1875–1941) and Giuseppe Vitali (1875–1932) at the beginning of the 1900's, applications as well as generalizations and extensions followed each other during the past century giving measure and integration theory a fundamental role in mathematical analysis. Here we follow the approach of first introducing Lebesgue's measure and accordingly Lebesgue's integral. In Section 2.1.1 we collect the main results of the theory without proofs,1 and in the following sections we develop its basic features

3. Curves and Differential Forms

In this chapter we discuss notions such as force, work, vector field, differential form, conservative vector field and its potential, and the solvability in an open set Ω R_n of the equation

$${\rm grad} \, U = F$$

We shall see that the vector field F is conservative, i.e., the equation grad U = F is solvable, if and only if the work along closed curves in Ω is zero, and we shall discuss how to compute a solution, a potential.

When n = 3, every function U of class C² satisfies the equation rot grad U = 0. Therefore, rot F = 0 in Ω is a necessary condition in order for the vector field F ∈ C¹ to be conservative in Ω. In terms of differential forms, we shall also see that rot F = 0 suffices for F to be conservative in simply connected domains.

Though Lebesgue’s theory of integration would allow us more general results, here we prefer to limit ourselves to the use of Riemann integral.

4. Holomorphic Functions

The theory of functions of one complex variable is one of the most central and fascinating chapters of mathematics. It has its prehistory with the works of Leonhard Euler (1707–1783), Joseph-Louis Lagrange (1736–1813), and Carl Friedrich Gauss (1777–1855), its gold period with Augustin-Louis Cauchy (1789–1857), G. F. Bernhard Riemann (1826–1866), Hermann Schwarz (1843–1921), and Karl Weierstrass (1815–1897), and it is the result of the contributions of many mathematicians in the period 1800–1950. The ideas, the methods, and the results of the theory of holomorphic functions play a fundamental role in several fields of mathematics both pure and applied, beyond their essential beauty. Here we shall limit ourselves to an elementary introduction

5. Surfaces and Level Sets

In the first two sections of this chapter we discuss the notion of surface and, related to it, the inverse and the implicit function theorems. Applications as well as some aspects of the local theory of surfaces will be discussed in the last two sections.

6. Systems of Ordinary Differential Equations

In this chapter we discuss a selection of classic results from the basic theory of ODE with the partial motivations of illustrating structures and techniques we have introduced. Of course, we refrain from any attempt of completeness and systematicity both for reasons of space and because this would lead into the theory of ODE and the theory of dynamical systems that have their autonomous development

Backmatter

Titel: Mathematical Analysis
verfasst von: Mariano Giaquinta
Giuseppe Modica
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4612-7
Print ISBN: 978-0-8176-4509-0
DOI: https://doi.org/10.1007/978-0-8176-4612-7