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The purpose of the volume is to provide a support textbook for a second lecture course on Mathematical Analysis. The contents are organised to suit, in particular, students of Engineering, Computer Science and Physics, all areas in which mathematical tools play a crucial role. The basic notions and methods concerning integral and differential calculus for multivariable functions, series of functions and ordinary differential equations are presented in a manner that elicits critical reading and prompts a hands-on approach to concrete applications. The pedagogical layout echoes the one used in the companion text Mathematical Analysis I. The book’s structure has a specifically-designed modular nature, which allows for great flexibility in the preparation of a lecture course on Mathematical Analysis. The style privileges clarity in the exposition and a linear progression through the theory. The material is organised on two levels. The first, reflected in this book, allows students to grasp the essential ideas, familiarise with the corresponding key techniques and find the proofs of the main results. The second level enables the strongly motivated reader to explore further into the subject, by studying also the material contained in the appendices. Definitions are enriched by many examples, which illustrate the properties discussed. A host of solved exercises complete the text, at least half of which guide the reader to the solution. This new edition features additional material with the aim of matching the widest range of educational choices for a second course of Mathematical Analysis.

Inhaltsverzeichnis

1. Numerical series

Abstract
This is the first of three chapters dedicated to series. A series formalises the idea of adding infinitely many terms of a sequence which can involve numbers (numerical series) or functions (series of functions). Using series we can represent an irrational number by the sum of an infinite sequence of increasingly smaller rational numbers, for instance, or a continuous map by a sum of infinitely many piecewise-constant functions defined over intervals of decreasing size. Since the definition itself of series relies on the notion of limit of a sequence, the study of a series' behaviour requires all the instruments used for such limits.
Claudio Canuto, Anita Tabacco

2. Series of functions and power series

Abstract
The idea of approximating a function by a sequence of simple functions, or known ones, lies at the core of several mathematical techniques, both theoretical and practical. For instance, to prove that a differential equation has a solution one can construct recursively a sequence of approximating functions and show they converge to the required solution. At the same time, explicitly finding the values of such a solution may not be possible, not even by analytical methods, so one idea is to adopt numerical methods instead, which can furnish approximating functions with a particularly simple form, like piecewise polynomials. It becomes thus crucial to be able to decide when a sequence of maps generates a limit function, what sort of convergence towards the limit we have, and which features of the functions in the sequence are inherited by the limit. All this will be the content of the first part of this chapter.
Claudio Canuto, Anita Tabacco

3. Fourier series

Abstract
The sound of a guitar, the picture of a footballer on tv, the trail left by an oil tanker on the ocean, the sudden tremors of an earthquake are all examples of events, either natural or caused by man, that have to do with travelling-wave phenomena. Sound for instance arises from the swift change in air pressure described by pressure waves that move through space. The other examples can be understood similarly using propagating electromagnetic waves, water waves on the ocean's surface, and elastic waves within the ground, respectively.
Claudio Canuto, Anita Tabacco

4. Functions between Euclidean spaces

Abstract
This chapter sees the dawn of the study of multivariable and vector-valued functions, that is, maps between the Euclidean spaces ℝ n and ℝ m or subsets thereof, with one of n and m bigger than 1. Subsequent chapters treat the relative differential and integral calculus and constitute a large part of the course.
Claudio Canuto, Anita Tabacco

5. Differential calculus for scalar functions

Abstract
While the previous chapter dealt with the continuity of multivariable functions and their limits, the next three are dedicated to differential calculus. We start in this chapter by discussing scalar functions.
Claudio Canuto, Anita Tabacco

6. Differential calculus for vector-valued functions

Abstract
In resuming the study of vector-valued functions started in Chapter 4, we begin by the various definitions concerning differentiability and introduce the Jacobian matrix, which gathers the gradients of the function's components, and the basic differential operators of order one and two. Then we will present the tools of differential calculus; among them, the so-called chain rule for differentiating composite maps has a prominent role, for it lies at the core of the idea of coordinate-system changes. After discussing the general theory, we examine in detail the special, but of the foremost importance, frame systems of polar, cylindrical, and spherical coordinates.
Claudio Canuto, Anita Tabacco

7. Applying differential calculus

Abstract
We conclude with this chapter the treatise of differential calculus for multivariate and vector-values functions. Two are the themes of concern: the Implicit Function Theorem with its applications, and the study of constrained extrema.
Claudio Canuto, Anita Tabacco

8. Integral calculus in several variables

Abstract
The definite integral of a function of one real variable allowed us, in Vol. I, to define and calculate the area of a sufficiently regular region in the plane. The present chapter extends this notion to multivariable maps by discussing multiple integrals; in particular, we introduce double integrals for dimension 2 and triple integrals for dimension 3. These new tools rely on the notions of a measurable subset of ℝ n and the corresponding n-dimensional measure; the latter extends the idea of the area of a plane region (n = 2), and the volume of a solid (n = 3) to more general situations.
Claudio Canuto, Anita Tabacco

9. Integral calculus on curves and surfaces

Abstract
With this chapter we conclude the study of multivariable integral calculus. In the first part we define integrals along curves in ℝ m and over surfaces in space, by considering first real-valued maps, then vector-valued functions. Integrating a vector field's tangential component along a curve, or its normal component on a surface, defines line and flux integrals respectively; these are interpreted in Physics as the work done by a force along a path, or the flow across a membrane immersed in a fluid. Curvilinear integrals rely, de facto, on integrals over real intervals, in the same way as surface integrals are computed by integrating over domains in the plane. A certain attention is devoted to how integrals depend upon the parametrisations and orientations of the manifolds involved.
Claudio Canuto, Anita Tabacco

10. Ordinary differential equations

Abstract
Differential equations are among the mathematical instruments most widely used in applications to other fields. The book's final chapter presents in a self-contained way the main notions, theorems and techniques of so-called ordinary differential equations. After explaining the basic principles underlying the theory we review the essential methods for solving several types of equations. We tackle existence and uniqueness issues for a solution to the initial value problem of a vectorial equation, then describe the structure of solutions of linear systems, for which algebraic methods play a central role, and equations of order higher than one. In the end the reader will find a concise introduction to asymptotic stability, with applications to pendulum motion, with and without damping.
Claudio Canuto, Anita Tabacco

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