A Course of Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
Published on 6. June 2016
508 pages
978-1-4831-3732-2 (ISBN)
Description
A Course of Mathematical Analysis, Part I is a textbook that shows the procedure for carrying out the various operations of mathematical analysis. Propositions are given with a precise statement of the conditions in which they hold, along with complete proofs. Topics covered include the concept of function and methods of specifying functions, as well as limits, derivatives, and differentials. Definite and indefinite integrals, curves, and numerical, functional, and power series are also discussed.
This book is comprised of nine chapters and begins with an overview of mathematical analysis and its meaning, together with some historical notes and the geometrical interpretation of numbers. The reader is then introduced to functions and methods of specifying them; notation for and classification of functions; and elementary investigation of functions. Subsequent chapters focus on limits and rules for passage to the limit; the concepts of derivatives and differentials in differential calculus; definite and indefinite integrals and applications of integrals; and numerical, functional, and power series.
This monograph will be a valuable resource for engineers, mathematicians, and students of engineering and mathematics.
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Content
Preface to the Seventh EditionIntroduction 1. Mathematical Analysis and Its Meaning 1. "Elementary" and "Higher" Mathematics 2. Magnitudes. Variables and Functional Relationships 3. Mathematical Analysis and Reality 2. Some Historical Notes 4. Great Russian Mathematicians: L. P. Euler, N. I . Lobachevskii, P. L. Chebyshev 5. Leading Russian Applied Mathematicians: N. E. Zhukovskii, S. A. Chaplygin, A. N. Krylov 3. Real Numbers 6. Real Numbers. The Real Axis 7. Intervals. Absolute Values 8. a Note on ApproximationsChapter I Functions 1. Functions and Methods of Specifying Them 9. The Concept of Function 10. Methods of Specifying Functions 2. Notation for and Classification of Functions 11. Notation 12. Function of a Function. Elementary Functions 13. The Classification of Functions 3. Elementary Investigation of Functions 14. Domain of Definition of a Function. Domain of Definiteness of an Analytic Expression 15. Elements of the Behavior of Functions 16. Graphical Investigation of a Function. Linear Combinations of Functions 4. Elementary Functions 17. Direct Proportionality and Linear Functions. Increments 18. Quadratic Functions 19. Inverse Proportionality and Linear Rational Functions 5. Inverse Functions. Power, Exponential and Logarithmic Functions 20. The Concept of Inverse Function 21. Power Functions 22. Exponential and Hyperbolic Functions 23. Logarithmic Functions 6. Trigonometric and Inverse Trigonometric Functions 24. Trigonometric Functions 25. Simple and Compound Harmonic Vibrations 26. Inverse Trigonometric FunctionsChapter II Limits 1. Basic Definitions 27. The Limit of a Function of an Integral Argument 28. Examples 29. The Limit of a Function of a Continuous Argument 2. Non-Finite Magnitudes. Rules for Passage to the Limit 30. Infinitely Large Magnitudes. Bounded Functions 31. Infinitesimals 32. Rules for Passage to the Limit 33. Examples 34. Tests for the Existence of a Limit 3. Continuous Functions 35. Continuity of a Function 36. Points of Discontinuity of a Function 37. General Properties of Continuous Functions 38. Operations on Continuous Functions. Continuity of the Elementary Functions 4. Comparison of Infinitesimals. Some Important Limits 39. Comparison of Infinitesimals. Equivalent Infinitesimals 40. Examples of Ratios of Infinitesimals 41. The Number e. Natural LogarithmsChapter III Derivatives and Differentials. The Differential Calculus 1. The Concept of Derivative. Rate of Change of a Function 42. Some Physical Concepts 43. Derivative of a Function 44. Geometrical Interpretation of Derivative 45. Some Properties of the Parabola 2. Differentiation of Functions 46. Differentiation of the Results of Arithmetical Operations 47. Differentiation of a Function of a Function 48. Derivatives of the Basic Elementary Functions 49. Logarithmic Differentiation. Differentiation of Inverse and Implicit Functions 50. Graphical Differentiation 3. Differentials. Differentiability of a Function 51. Differentials and Their Geometrical Interpretation 52. Properties of the Differential 53. Application of the Differential to Approximations 54. Differentiability of a Function. Smoothness of a Curve 4. Derivative as Rate of Change (Further Examples) 55. Rate of Change of a Function with Respect to a Function. Parametric Specification of Functions and Curves 56. Rate of Change of Radius Vector 57. Rate of Change of Length of Arc 58.
