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Outline Course of Pure Mathematics
- 1st Edition - January 1, 1968
- Author: A. F. Horadam
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 4 7 9 0 - 1
Outline Course of Pure Mathematics presents a unified treatment of the algebra, geometry, and calculus that are considered fundamental for the foundation of undergraduate… Read more
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Request a sales quoteOutline 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.
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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
Index
- No. of pages: 594
- Language: English
- Edition: 1
- Published: January 1, 1968
- Imprint: Pergamon
- eBook ISBN: 9781483147901
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