A Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 1
Published on 16. May 2014
558 pages
978-1-4831-5636-1 (ISBN)
Description
A Course of Higher Mathematics, I: Elementary Calculus is a five-volume course of higher mathematics used by mathematicians, physicists, and engineers in the U.S.S.R. This volume deals with calculus and principles of mathematical analysis including topics on functions of single and multiple variables. The functional relationships, theory of limits, and the concept of differentiation, whether as theories and applications, are discussed. This book also examines the applications of differential calculus to geometry. For example, the equations to determine the differential of arc or the parameters of a curve are shown. This text then notes the basic problems involving integral calculus, particularly regarding indefinite integrals and their properties. The application of definite integrals in the calculation of area of a sector, the length of arc, and the calculation of the volumes of solids of a given cross-section are explained. This book further discusses the basic theory of infinite series, applications to approximate evaluations, Taylor's formula, and its extension. Finally, the geometrical approach to the concept of a number is reviewed. This text is suitable for physicists, engineers, mathematicians, and students in higher mathematics.
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Content
IntroductionPrefaces to Eighth and Sixteenth Russian EditionsChapter I. Functional Relationships and the Theory of Limits § 1. Variables 1. Magnitude and Its Measurement 2. Number 3. Constants and VariableS 4. Interval 5. The concept of function 6. The Analytic Method of Representing Functional Relationships 7. Implicit Functions 8. The Tabular Method 9. The Graphical Method of Representing Numbers 10. Coordinates 11. Graphs. The Equation of a Curve 12. Linear Functions 13. Increment. The Basic Property of a Linear Function 14. Graph of Uniform Motion 15. Empirical Formula 16. Parabola of the Second Degree 17. Parabola of the Third Degree 18. The Law of Inverse Proportionality 19. Power Functions 20. Inverse Functions 21. Many-valued Functions 22. Exponential and Logarithmic Functions 23. Trigonometric Functions 24. Inverse Trigonometric, or Circular, Functions § 2. The Theory of Limits. Continuous Functions 25. Ordered Variables 26. Infinitesimals 27. The Limit of a Variable 28. Basic Theorems 29. Infinitely Large Magnitudes 30. Monotonic Variables 31. Cauchy's Test for the Existence of a Limit 32. Simultaneous Variation of Two Variables, Connected by a Functional Relationship 33. Example 34. Continuity of Functions 35. The Properties of Continuous Functions 36. Comparison of Infinitesimals and of Infinitely Large Magnitudes 37. Examples 38. The Number E 39. Unproved Hypotheses 40. Real Numbers 41. The Operations on Real Numbers 42. The Strict Bounds of Numerical Sets. Tests for the Existence of a Limit 43. Properties of Continuous Functions 44. Continuity of Elementary Functions ExercisesChapter II. Differentiation: Theory and Applications § 3. Derivatives and Differentials of the First Order 45. The Concept of Derivative 46. Geometrical Significance of the Derivative 47. Derivatives of some Simple Functions 48. Derivatives of Functions of a Function, and of Inverse Functions 49.Table of Derivatives, and Examples 50. The Concept of Differential 51. Some Differential Equations 52. Estimation of Errors § 4. Derivatives and Differentials of Higher Orders 53. Derivatives of Higher Orders 54. Mechanical Significance of the Second Derivative 55. Differentials of Higher Orders 56. Finite Differences of Functions § 5· Application of Derivatives to the Study of Functions 57. Tests for Increasing and Decreasing Functions 58. Maxima and Minima of Functions 59. Curve Tracing 60. The Greatest and Least Values of a Function 61. Format's Theorem 62. Rolle's Theorem 63. Lagrange's Formula 64. Cauchy's Formula 65. Evaluating Indeterminate Forms 66. Other Indeterminate Forms § 6· Functions of Two Variables 67. Basic Concepts 68. The Partial Derivatives and Total Differential of a Function of two Independent Variables 69. Derivatives of Functions of a Function and of Implicit Functions § 7· Some Geometrical Applications of the Differential Calculus 70. The Differential of Arc 71. Concavity, Convexity, and Curvature 72. Asymptotes 73. Curve-tracing 74. The Parameters of a Curve 75. Van der Waal's Equation 76. Singular Points of Curves 77. Elements of a Curve 78. The Catenary 79. The Cycloid 80. Epicycloid and Hypocycloid 81. Involute of a Circle 82.
