Outline Course of Pure Mathematics - 1st Edition - ISBN: 9780080125930, 9781483147901

Outline Course of Pure Mathematics

1st Edition

Authors: A. F. Horadam
eBook ISBN: 9781483147901
Imprint: Pergamon
Published Date: 1st January 1968
Page Count: 594
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Outline Course of Pure Mathematics presents a unified treatment of the algebra, geometry, and calculus that are considered fundamental for the foundation of undergraduate mathematics. This book discusses several topics, including elementary treatments of the real number system, simple harmonic motion, Hooke's law, parabolic motion under gravity, sequences and series, polynomials, binomial theorem, and theory of probability. Organized into 23 chapters, this book begins with an overview of the fundamental concepts of differential and integral calculus, which are complementary processes for solving problems of the physical world. This text then explains the concept of the inverse of a function that is a natural complement of the function concept and introduces a convenient notation. Other chapters illustrate the concepts of continuity and discontinuity at the origin. This book discusses as well the significance of logarithm and exponential functions in scientific and technological contexts. This book is a valuable resource for undergraduates and advanced secondary school students.

Table of Contents

Greek Alphabet

Select Bibliography

Chapter 1. Differential Calculus

1. Differentiation (Revision)

2. Differentials

3. Maxima and Minima (Revision)

Exercises 1

Chapter 2. Inverse Trigonometrical Functions

4. Nature of Inverse Functions

5. Special Properties of Inverse Trigonometrical Functions

Exercises 2

Chapter 3. Elementary Analysis

6. Limits (Revision). The Symbol

7. Concept of the Limit of a Function

8. Concept of Continuity

9. The Mean Value Theorem. Rolle's Theorem

10. l'Hospital's Rule

Exercises 3

Chapter 4. Expotential and Logarithmic Functions

11. Exponential Function. Exponential Number

12. Graphs of the Exponential and Logarithmic Functions

13. Differentiation of the Exponential and Logarithmic Functions

Exercises 4

Chapter 5. Hyperbolic Functions

14. The Hyperbolic Functions

15. Differentiation of the Hyperbolic Functions

16. Graphs of the Hyperbolic Functions

17. Inverse Hyperbolic Functions

18. The Gudermannian and Inverse Gudermannian

Exercises 5

Chapter 6. Partial Differentiation

19. n-Dimensional Geometry

20. Polar Coordinates

21. Partial Differentiation

22. Total Differentials

Exercises 6

Chapter 7. Indefinite Integrals

23. The Indefinite Integral

24. Standard Integrals

25. Techniques of Integration: Change of Variable (Substitution, Transformation)

26. Techniques of Integration: Trigonometric Denominator

27. Techniques of Integration: Integration by Parts

28. Techniques of Integration: Partial Fractions

29. Techniques of Integration: Quadratic Denominator

Exercises 7

Chapter 8. Definite Integrals

30. Elementary First-order Differential Equations (Method of Separation of Variables)

31. The Definite Integral

32. Improper Integrals

33. The Definite Integral as an Area and as the Limit of a Sum

34. Properties of f (x) dx

35. Reduction Formula

36. An Integral Approach to the Theory of Logarithmic Functions

Exercises 8

Chapter 9. Infinite Series and Sequences

37. Sequences

38. Convergence and Divergence of Infinite Series

39. Tests for Convergence

40. Alternating Series. Absolute and Conditional Convergence

41. Maclaurin's Series

42. Leibniz's Formula

Exercises 9

Chapter 10. Complex Numbers

43. The Real Number System

44. Number Rings and Fields

45. Intuitive Approach to Complex Numbers

46. Formal Development of Complex Numbers

47. Geometrical Representation of Complex Numbers. The Argand Diagram

48. Euler's Theorem (1742)

49. Complex Numbers and Polynomial Equations

50. Elementary Symmetric Functions

51. Some Typical Problems Involving Complex Numbers

52. Hypercomplex Numbers (Quaternions)

Exercises 10

Chapter 11. Matrices

53. Linear Transformations and Matrices

54. Formal Definitions

55. Matrices and Vectors

56. Matrices and Linear Equations

57. Matrices and Determinants

Exercises 11

Chapter 12. Determinants

58. Formal Definitions and Basic Properties

59. Minors and Cofactors. Expansion of a Determinant

60. Adjoint Determinant

61. Inverse of a Matrix

62. Solution of Simultaneous Linear Equations

63. Elimination and Eigenvalues

64. Determinants and Vectors

Exercises 12

Chapter 13. Sets and Their Applications. Boolean Algebra

65. The Language of Set Theory

66. Transfinite Numbers

67. Venn Diagrams

68. Boolean Algebra and Sets

69. Number of Elements in a Set

Exercises 13

Chapter 14. Groups

70. Intuitive Approach to Groups

71. Formal Definitions and Basic Properties

72. Survey of Groups of Order 2, 3, 4, 5, 6

73. Concepts of Subgroup and Generators

74. Isomorphism

75. Typical Problems in Elementary Group Theory 2

76. Abstract Rings and Fields

Exercises 14

Chapter 15. The Nature of Geometry

77. The Problem of Parallelism. Elements at Infinity

78. Homogeneous Co-ordinates. Circular Points at Infinity

79. Euclidean Group. Projective Geometry

80. Cross-ratio

Exercises 15

Chapter 16. Conics

81. Conics as Plane Loci and as Conic Sections

82. General Equation of a Conic

83. Standard Equations of the Conics

84. Conics and the Line at Infinity

85. Quadratic Equation Representing a Line-Pair

86. Tangent at a Given Point

87. Elementary Theory of Pole and Polar

88. Reduction of a Central Conic to Standard Form

Exercises 16

Chapter 17. The Parabola

89. Basic Properties (Revision Summary)

90. Selected Problems Solved Parametrically

91. Normals to a Parabola

Exercises 17

Chapter 18. The Ellipse

92. Basic Properties

93. Selected Problems Solved Parametrically

94. Conjugate Diameters

Exercises 18

Chapter 19. The Hyperbola

95. Basic Properties

96. Asymptotes

97. Rectangular Hyperbola

Exercises 19

Chapter 20. CurvesS: Cartesian Coordinates

98. Concavity, Convexity, Point of Inflexion

99. Some Rules for Curve-sketching

100. The Problem of Asymptotes

101. Double Points

102. Selected Examples of Curve-sketching in Cartesian Coordinates

103. Composition of Curves

104. Families of Curves

105. Special Higher Plane Curves

106. Parametric Curves

Exercises 20

Chapter 21. Curvature

107. Intrinsic Coordinates

108. Curvature

109. Radius of Curvature

Exercises 21

Chapter 22. Curves: Polar Coordinates

110. Equations of Line and Circle in Polar Coordinates

111. Equations of Conics in Polar Coordinates

112. Some Rules for Curve-sketching

113. Selected Examples of Curve-sketching in Polar Coordinates

114. Spirals and Rose Curves

115. Meaning of r dq/dr

116. Tangent in Polar Coordinates

117. Equiangular Spiral

Exercises 22

Chapter 23. Geometrical Applications of the Definite Integral

118. Area in Polar Coordinates

119. Volume of a Solid of Revolution

120. Length of a Curve

121. Surface Area of a Solid of Revolution

122. Approximate (Numerical) Integration. Concluding Remarks

Exercises 23

New Horizons

Solutions to Exercises



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© Pergamon 1968
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About the Author

A. F. Horadam