Mathematical Analysis. Differential Calculus.

DESPRE CARTE

Scrisa intr-un limbaj elevat, dar pe deplin accesibila, presarata cu sute de exemple care lamuresc materialul teoretic al acestei discipline fundamentale, raspunzand si cerintelor actuale ale invatarii prin exemple, cartea se constituie intr-un instrument deosebit de util pentru studentii unei universitati de prestigiu.

Referent stiintific – Prof.dr. Pasc Gavruta

Materialul tratat inglobeaza armonios si logic aspectele diferentiale ale analizei matematice, iar multitudinea de probleme rezolvate ori propuse spre rezolvare fac deschiderea catre aplicatiile practice ale analizei.

Este o lucrare reusita, atat din punct de vedere pedagogic, cat si din punct de vedere stiintific, imbinand astfel criteriile utilitatii si eficientei.

Referent stiintific – Prof.dr. Octavian Lipovan

CONTENTS

Preface 5

PART I. ANALYSIS OF FUNCTIONS OF A SINGLE REAL VARIABLE
1. NUMERICAL SEQUENCES AND SERIES 9
1.1. Sequences 9
1.2. Definitions. General Criteria 22
1.3. Series with Nonegative Terms 32
1.4. Approximate Computation of Sums 40
1.5. Improper Integrals and Series 43
1.6. Infinite Products 58
1.7. Solved Problems 62
1.8. Exercises 76

2. APPROXIMATIONS WITH TAYLOR POLYNOMIALS 108
2.1. Derivatives 108
2.2. Differentiability 110
2.3. Taylor`s Formula 113
2.4. Approximation of Functions 117
2.5. Finding the Extrema 122
2.6. Algebraic Application 125
2.7. Solved Problems 127
2.8. Exercises 133

3. SEQUENCES AND SERIES OF FUNCTIONS 144
3.1. Convergence of Sequences of Functions 144
3.2. Series of Functions 150
3.3. Power Series 159
3.4. Taylor Series 164
3.5. Other Polynomial Approximations 168
3.6. Fourier Series 172
3.7. Solved Problems 182
3.8. Exercises 205

4. METRIC SPACES. FIXED POINT THEOREMS 228
4.1. Metrics, Norms, Inner-products 228
4.2. Elements of Topolgy 233
4.3. Sequences 237
4.4. Fixed Point Theorems 240
4.5. Solved Problems 246
4.6. Exercises 253

PART II. ANALYSIS OF FUNCTIONS OF SEVERAL REAL VARIABLES
5. LIMIT AND CONTINUITY IN IRp 260
5.1. Limits of Sequences 260
5.2. Vectorial Functions 272
5.3. Limit of a Function 278
5.4. Continuous Functions 287
5.5. Solved Problems 297
5.6. Exercises 306

6. DIFFERENTIABLE MAPPINGS 318
6.1. Partial Derivatives 318
6.2. Directional Derivatives 326
6.3. Differentiability 332
6.4. The Differentials and Derivatives of Composite Functions 346
6.5. Homogenous Functions 351
6.6. Differentials of Higher Order 355
6.7. Elements of Calculus in the Theory of Fields 360
6.8. Solved Problems 369
6.9. Exercises 381

7. DIFFERENTIABLE ISOMORPHISMS 391
7.1. Regular Transformations 391
7.2. Implicit Functions 396
7.3. Functional Dependence 406
7.4. Changes of Variables 413
7.5. Solved Problems 426
7.6. Exercises 434

8. POLYNOMIAL APPROXIMATIONS 444
8.1. Taylor`s Formula 444
8.2. Local Extrema 452
8.3. Conditional Extrema 464
8.4. Solved Problems 473
8.5. Exercises 478

BIBLIOGRAPHY 487

INDEX 488