Mathematical analysis I / Claudio Canuto, Anita Tabacco.

Format:

Book

Author/Creator:

Canuto, C.

Contributor:

Tabacco Vignati, Anita Maria

Series:

Universitext

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

xii, 433 pages : illustrations (some color) ; 24 cm.

Place of Publication:

Milan ; New York : Springer, [2008]

Summary:

The purpose of the volume is to provide a support for a first course in Mathematical Analysis, along the lines of the recent Programme Specifications for mathematical teaching in European universities. The contents are organised to appeal especially to Engineering, Physics and Computer Science students, all areas in which mathematical tools play a crucial role. Basic notions and methods of differential and integral calculus for functions of one real variable are presented in a manner that elicits critical reading and prompts a hands-on approach to concrete applications. The layout has a specifically-designed modular nature, allowing the instructor to make flexible didactical choices when planning an introductory lecture course. The book may in fact be employed at three levels of depth. At the elementary level the student is supposed to grasp the very essential ideas and familiarise with the corresponding key techniques. Proofs to the main results befit the intermediate level, together with several remarks and complementary notes enhancing the treatise. The last, and farthest-reaching, level consists of links to online material, which enable the strongly motivated reader to explore further into the subject. Definitions and properties are furnished with substantial examples to stimulate the learning process. Over 350 solved exercises complete the text, at least half of which guide the reader to the solution. Written for: Engineering, Physics and Computer Science students.

Contents:

1 Basic notions 1

1.1 Sets 1

1.2 Elements of mathematical logic 5

1.2.1 Connectives 5

1.2.2 Predicates 6

1.2.3 Quantifiers 7

1.3 Sets of numbers 8

1.3.1 The ordering of real numbers 12

1.3.2 Completeness of R 17

1.4 Factorials and binomial coefficients 18

1.5 Cartesian product 21

1.6 Relations in the plane 23

2 Functions 31

2.1 Definitions and first examples 31

2.2 Range and pre-image 36

2.3 Surjective and injective functions; inverse function 38

2.4 Monotone functions 41

2.5 Composition of functions 43

2.5.1 Translations, rescalings, reflections 45

2.6 Elementary functions and properties 47

2.6.1 Powers 48

2.6.2 Polynomial and rational functions 50

2.6.3 Exponential and logarithmic functions 50

2.6.4 Trigonometric functions and inverses 51

3 Limits and continuity I 65

3.1 Neighbourhoods 65

3.2 Limit of a sequence 66

3.3 Limits of functions; continuity 72

3.3.1 Limits at infinity 72

3.3.2 Continuity. Limits at real points 74

3.3.3 One-sided limits; points of discontinuity 82

3.3.4 Limits of monotone functions 85

4 Limits and continuity II 89

4.1 Theorems on limits 89

4.1.1 Uniqueness and sign of the limit 89

4.1.2 Comparison theorems 91

4.1.3 Algebra of limits. Indeterminate forms of algebraic type 96

4.1.4 Substitution theorem 102

4.2 More fundamental limits. Indeterminate forms of exponential type 105

4.3 Global features of continuous maps 108

5 Local comparison of functions. Numerical sequences and series 123

5.1 Landau symbols 123

5.2 Infinitesimal and infinite functions 130

5.3 Asymptotes 135

5.4 Further properties of sequences 137

5.5 Numerical series 141

5.5.1 Positive-term series 146

5.5.2 Alternating series 149

6 Differential calculus 167

6.1 The derivative 167

6.2 Derivatives of the elementary functions. Rules of differentiation 170

6.3 Where differentiability fails 175

6.4 Extrema and critical points 178

6.5 Theorems of Rolle and of the Mean Value 181

6.6 First and second finite increment formulas 183

6.7 Monotone maps 185

6.8 Higher-order derivatives 187

6.9 Convexity and inflection points 189

6.9.1 Extension of the notion of convexity 192

6.10 Qualitative study of a function 193

6.10.1 Hyperbolic functions 195

6.11 The Theorem of de l'Hopital 197

6.11.1 Applications of de l'Hopital's theorem 199

7 Taylor expansions and applications 223

7.1 Taylor formulas 223

7.2 Expanding the elementary functions 227

7.3 Operations on Taylor expansions 234

7.4 Local behaviour of a map via its Taylor expansion 242

8 Geometry in the plane and in space 257

8.1 Polar, cylindrical, and spherical coordinates 257

8.2 Vectors in the plane and in space 260

8.2.1 Position vectors 260

8.2.2 Norm and scalar product 263

8.2.3 General vectors 268

8.3 Complex numbers 269

8.3.1 Algebraic operations 270

8.3.2 Cartesian coordinates 271

8.3.3 Trigonometric and exponential form 273

8.3.4 Powers and nth roots 275

8.3.5 Algebraic equations 277

8.4 Curves in the plane and in space 279

8.5 Functions of several variables 284

8.5.1 Continuity 284

8.5.2 Partial derivatives and gradient 286

9 Integral calculus I 299

9.1 Primitive functions and indefinite integrals 300

9.2 Rules of indefinite integration 304

9.2.1 Integrating rational maps 310

9.3 Definite integrals 317

9.4 The Cauchy integral 318

9.5 The Riemann integral 320

9.6 Properties of definite integrals 326

9.7 Integral mean value 328

9.8 The Fundamental Theorem of integral calculus 331

9.9 Rules of definite integration 335

9.9.1 Application: computation of areas 337

10 Integral calculus II 355

10.1 Improper integrals 355

10.1.1 Unbounded domains of integration 355

10.1.2 Unbounded integrands 363

10.2 More improper integrals 367

10.3 Integrals along curves 368

10.3.1 Length of a curve and arc length 373

10.4 Integral vector calculus 376

11 Ordinary differential equations 387

11.1 General definitions 387

11.2 First order differential equations 388

11.2.1 Equations with separable variables 392

11.2.2 Linear equations 394

11.2.3 Homogeneous equations 397

11.2.4 Second order equations reducible to first order 398

11.3 Initial value problems for equations of the first order 399

11.3.1 Lipschitz functions 399

11.3.2 A criterion for solving initial value problems 402

11.4 Linear second order equations with constant coefficients 404.

Notes:

Includes index.

ISBN:

9788847008755

8847008751

OCLC:

230991574