Mathematical Analysis

S. C. Malik, Savita Arora

New Age International, 1992 - Mathematics - 903 pages

The Book Is Intended To Serve As A Text In Analysis By The Honours And Post-Graduate Students Of The Various Universities. Professional Or Those Preparing For Competitive Examinations Will Also Find This Book Useful.The Book Discusses The Theory From Its Very Beginning. The Foundations Have Been Laid Very Carefully And The Treatment Is Rigorous And On Modem Lines. It Opens With A Brief Outline Of The Essential Properties Of Rational Numbers And Using Dedekinds Cut, The Properties Of Real Numbers Are Established. This Foundation Supports The Subsequent Chapters: Topological Frame Work Real Sequences And Series, Continuity Differentiation, Functions Of Several Variables, Elementary And Implicit Functions, Riemann And Riemann-Stieltjes Integrals, Lebesgue Integrals, Surface, Double And Triple Integrals Are Discussed In Detail. Uniform Convergence, Power Series, Fourier Series, Improper Integrals Have Been Presented In As Simple And Lucid Manner As Possible And Fairly Large Number Solved Examples To Illustrate Various Types Have Been Introduced.As Per Need, In The Present Set Up, A Chapter On Metric Spaces Discussing Completeness, Compactness And Connectedness Of The Spaces Has Been Added. Finally Two Appendices Discussing Beta-Gamma Functions, And Cantors Theory Of Real Numbers Add Glory To The Contents Of The Book.

Contents

• REAL NUMBERS: 1
• OPEN SETS CLOSED SETS: 33
• Closure of a Set: 42
• Countable and Uncountable Sets: 49
• INFINITE SERIES: 109
• Positive Term Series: 114
• Comparison Tests for Positive Term Series: 118
• Cauchys Root Test: 124
• DAlemberts Ratio Test: 125
• Raabes Test: 127
• Logarithmic Test: 131
• Integral Test: 132
• Gausss Test: 135
• Series with Arbitrary Terms: 139
• Rearrangement of Terms: 148
• FUNCTIONS OF A SINGLE VARIABLE I: 154
• Continuous Functions: 165
• Functions Continuous on Closed Intervals: 174
• Uniform Continuity: 179
• FUNCTIONS OF A SINGLE VARIABLE II: 185
• Continuous Functions: 188
• Increasing and Decreasing Functions: 191
• Darbouxs Theorem: 194
• Rolles Theorem: 195
• Lagranges Mean Value Theorem: 196
• Cauchys Mean Value Theorem: 198
• Higher Order Derivatives: 206
• APPLICATIONS OF TAYLORS THEOREM: 216
• Indeterminate Forms: 223
• FUNCTIONS: 236
• Exponential Functions: 238
• Logarithmic Functions: 240
• Trigonometric Functions: 243
• Functional Equations: 249
• Functions of Bounded Variation: 251
• VectorValued Functions: 262
• THE RIEMANN INTEGRAL: 270
• Refinement of Partitions: 277
• Integrability of the Sum and Difference: 284
• The Integral as a Limit of Sums: 293
• Some Integrable Functions: 300
• The Fundamental Theorem of Calculus: 306
• Integration by Parts: 316
• THE RIEMANNSTIELTJES INTEGRAL: 330
• IMPROPER INTEGRALS: 351
• Infinite Range of Integration: 370
• Integrand as a Product of Functions: 389
• UNIFORM CONVERGENCE: 404
• POWER SERIES 1 Generic Term: 446
• Properties of Functions Expressible as Power Series: 450
• Abels Theorem: 453
• FOURIER SERIES 1 Trigonometrical Series: 463
• Some Preliminary Theorems: 465
• The Main Theorem: 471
• Intervals Other Than π: 479
• FUNCTIONS OF SEVERAL VARIABLES 1 Explicit and Implicit Functions: 492
• Continuity: 501
• Partial Derivatives: 505
• 4: 507
• Differentiability: 509
• Partial Derivatives of Higher Order: 517
• Differentials of Higher Order: 524
• Functions of Functions: 526
• Change of Variables: 533
• Taylors Theorem: 544
• Maxima and Minima: 548
• Functions of Several Variables: 554
• IMPLICIT FUNCTIONS 1 Definition: 562
• Jacobians: 567
• Stationary Values under Subsidiary Conditions: 575
• INTEGRATION ON R2 1 Line Integrals: 588
• Double Integrals: 596
• Double Integrals Over a Region: 618
• Greens Theorem: 629
• Change of Variables: 637
• INTEGRATION ON R³ Rectifiable Curves: 652
• 2: 653
• Line Integrals: 657
• 4: 659
• Surface Integrals: 670
• Stokes Theorem First generalization of Greens Theorem: 687
• The Volume of a Cylindrical Solid by Double Integrals: 692
• Volume Integrals Triple Integrals: 698
• Gausss Theorem Divergence Theorem: 708
• METRIC SPACES 1 Definitions and Examples: 726
• Open and Closed Sets: 737
• Convergence and Completeness: 758
• Continuity and Uniform Continuity: 768
• Compactness: 781
• Connectedness: 800
• THE LEBESGUE INTEGRAL 1 Measurable Sets: 811
• Sets of Measure Zero: 820
• Borel Sets: 824
• Measurable Functions: 828
• Measurability of the Sum Difference Product and Quotient Measurable Functions: 831
• Lebesgue Integral: 836
• Properties of Lebesgue Integral for Bounded Measurable Functions: 839
• Lebesgue Integral of a Bounded Function Over a Set of Finite Measure: 845
• Lebesgue Integral for Unbounded Functions: 850
• The General Integral: 853
• Lebesgue Theorem on Bounded Convergence: 857
• Integrability and Measurability: 859
• Lebesgue Integral on Unbounded Sets or Intervals: 869
• BETA AND GAMMA FUNCTIONS: 872
• Order in R: 885
• 440: 897
• 492: 898
• 588: 899
• 404: 901

Bibliographic information

• Title: Mathematical Analysis
• Authors: S. C. Malik, Savita Arora
• Edition: illustrated
• Publisher: New Age International, 1992
• ISBN: 8122403239, 9788122403237
• Length: 903 pages
• Subjects: Mathematics / Mathematical Analysis