Horadam Outline Course of Pure Mathematics

1. Auflage 2014
ISBN: 978-1-4831-4790-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 594 Seiten, Web PDF

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.

Autoren/Hrsg.

Horadam, A. F.

Inhaltsverzeichnis

1;Front Cover
;1
2;Outline Course of Pure Mathematics
;4
3;Copyright Page
;5
4;Table of Contents
;8
5;PREFACE;12
6;GREEK ALPHABET;15
7;SELECT BIBLIOGRAPHY;16
8;CHAPTER 1. DIFFERENTIAL CALCULUS
;18
8.1;§ 1. DIFFERENTIATION (REVISION)
;18
8.2;§ 2. DIFFERENTIALS;24
8.3;§ 3. MAXIMA AND MINIMA (REVISION)
;25
8.4;EXERCISES 1;27
9;Chapter 2. INVERSE TRIGONOMETRICAL FUNCTIONS
;31
9.1;§ 4. NATURE OF INVERSE FUNCTIONS
;31
9.2;§ 5. SPECIAL PROPERTIES OF INVERSE TRIGONOMETRICAL FUNCTIONS
;33
9.3;EXERCISES 2;34
10;CHAPTER 3. ELEMENTARY ANALYSIS
;36
10.1;
6. LIMITS (REVISION). THE SYMBOL ;36
10.2;§ 7. CONCEPT OF THE LIMIT OF A FUNCTION
;38
10.3;§ 8. CONCEPT OF CONTINUITY
;39
10.4;§ 9. THE MEAN VALUE THEOREM. ROLLE'S THEOREM
;42
10.5;§ 10. L' H O S P I TA L' S RULE (1694) (L' Ho spital, 1661-1704, French
;44
10.6;EXERCISES 3;47
11;CHAPTER 4. EXPONENTIAL AND LOGARITHMIC FUNCTIONS
;48
11.1;§
11. EXPONENTIAL FUNCTION. EXPONENTIALNUMBER;48
11.2;§ 12. GRAPHS OF THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS
;52
11.3;§ 13. DIFFERENTIATION OF THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS
;54
11.4;EXERCISES 4;58
12;CHAPTER 5. HYPERBOLIC FUNCTIONS
;61
12.1;§ 14. THE HYPERBOLIC FUNCTIONS
;61
12.2;§ 15. DIFFERENTIATION OF THE HYPERBOLIC FUNCTIONS
;63
12.3;§ 1 6. GRAPHS OF THE HYPERBOLIC FUNCTIONS
;64
12.4;§ 17. INVERSE HYPERBOLIC FUNCTIONS
;68
12.5;§ 18. THE GUDERMANNIAN AND INVERSE GUDERMANNIAN
;68
12.6;EXERCISES 5;70
13;CHAPTER 6. PARTIAL DIFFERENTIATION
;73
13.1;§
19. n-DIMENSIONAL GEOMETRY;73
13.2;§ 20. POLAR COORDINATES;74
13.3;§ 21. PARTIAL DIFFERENTIATION;76
13.4;§ 22. TOTAL DIFFERENTIA
LS;82
13.5;EXERCISES 6;83
14;CHAPTER 7. INDEFINITE INTEGRALS
;86
14.1;§ 23. THE INDEFINITE INTEGRAL
;86
14.2;§ 24. STANDARD INTEGRALS;88
14.3;§ 25. TECHNIQUES OF INTEGRATION: CHANGE OF VARIABLE (SUBSTITUTION, TRANSFORMATION)
;89
14.4;§ 26. TECHNIQUES OF INTEGRATION: TRIGONOMETRIC DENOMINATOR
;95
14.5;§ 27. TECHNIQUES OF INTEGRATION: INTEGRATION BY PARTS;96
14.6;§ 28. TECHNIQUES OF INTEGRATION: PARTIAL FRACTIONS
;98
14.7;§ 29. TECHNIQUES OF INTEGRATION: QUADRATIC DENOMINATOR
;101
14.8;EXERCISES 7;102
15;CHAPTER 8. DEFINITE INTEGRALS
;104
15.1;§ 30. ELEMENTARY FIRST-ORDER DIFFERENTIAL EQUATIONS (METHOD OF SEPARATION OF VARIABLES)
;104
15.2;§ 31. THE DEFINITE INTEGRAL
;107
15.3;§ 3 2. IMPROPER INTEGRALS;109
15.4;§ 33. THE DEFINITE INTEGRAL AS AN AREA AND AS THE LIMIT OF A SUM
;111
15.5;§ 34. PROPERTIES OF f(x) dx
;117
15.6;§ 35. REDUCTION FORMULAE;120
15.7;§ 36. AN INTEGRAL APPROACH TO THE THEORY OF LOGARITHMIC FUNCTIONS
;125
15.8;EXERCISES 8;128
16;CHAPTER 9. INFINITE SERIES AND SEQUENCES
;135
16.1;§ 37. SEQUENCES
;135
16.2;§ 38. CONVERGENCE AND DIVERGENCE OF INFINITE SERIES
;136
16.3;§ 39. TESTS FOR CONVERGENCE
;138
16.4;§ 40. ALTERNATING SERIES. ABSOLUTE AND CONDITIONAL CONVERGENCE
;141
16.5;§ 41. MACLAURIN'S SERIES;142
16.6;§ 42. LEIBNIZ'S FORMULA;147
16.7;EXERCISES 9;151
17;CHAPTER 10. COMPLEX NUMBERS
;156
17.1;§ 43. THE REAL NUMBER SYSTEM
;156
17.2;§ 44. NUMBER RINGS AND FIELDS;159
17.3;§ 45. INTUITIVE APPROACH TO COMPLEX NUMBERS
;160
17.4;§ 46. FORMAL DEVELOPMENT OF COMPLEX NUMBERS
;162
17.5;§ 47. GEOMETRICAL REPRESENTATION OF COMPLEX NUMBERS. THE ARGAND DIAGRAM
;165
17.6;§ 48. EULER'S THEOREM (1742)
;170
17.7;§ 49. COMPLEX NUMBERS AND POLYNOMIAL EQUATIONS
;172
17.8;§ 50. ELEMENTARY SYMMETRIC FUNCTIONS
;173
17.9;§ 51. SO ME TYPICAL PROBLEMS INVOLVING COMPLEX NUMBERS
;175
17.10;§ 52. HYPERCOMPLEX NUMBERS (QUATERNIONS)
;182
17.11;EXERCISES 10;184
18;CHAPTER 11. MATRICES
;188
18.1;§ 5
3. LINEAR TRANSFORMATIONS AND MATRICES;189
18.2;§ 54. FORMAL DEFINITIONS;196
18.3;§ 55. MATRICES AND VECTORS;201
18.4;§ 56. MATRICES AND LINEAR EQUATIONS
;206
18.5;§ 57. MATRICES AND DETERMINANTS;208
18.6;EXERCISES 11;211
19;CHAPTER 12. DETERMINANTS
;215
19.1;§ 58. FORMAL DEFINITIONS AND BASIC PROPERTIES
;215
19.2;§ 59. MINORS AND COFACTORS. EXPANSION OF A DETERMINANT
;220
19.3;§ 60. ADJOINT DETERMINANT
;224
19.4;§61. INVERSE OF A MATRIX;225
19.5;§ 62. SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS
;226
19.6;§ 63. ELIMINATION AND EIGENVALUES;229
19.7;§ 64. DETERMINANTS AND VECTORS
;230
19.8;EXERCISES 12;235
20;CHAPTER 13. SETS AND THEIR APPLICATIONS. BOOLEAN ALGEBRA
;241
20.1;§ 65. THE LANGUAGE OF SET THEORY;241
20.2;§ 66. TRANSFINITE NUMBERS
;245
20.3;§ 67. VENN DIAGRAMS
;248
20.4;§ 68. BOOLEAN ALGEBRA AND SETS
;249
20.5;§ 69. NUMBER OF ELEMENTS IN A SET;255
20.6;EXERCISES 13;256
21;CHAPTER 14. GROUPS
;258
21.1;§ 70. INTUITIVE APPROACH TO GROUPS
;258
21.2;§ 71. FORMAL DEFINITIONS AND BASIC PROPERTIES
;260
21.3;§ 72. SURVEY OF GROUPS OF ORDER 2, 3,4, 5, 6
;262
21.4;§ 73. CONCEPTS OF SUBGROUP AND GENERATORS
;267
21.5;
74. ISOMORPHISM;267
21.6;§
75. TYPICAL PROBLEMS IN ELEMENTARY GROUPTHEORY;268
21.7;§ 76. ABSTRACT RINGS AND FIELDS;270
21.8;EXERCISES 14;273
22;CHAPTER 15. THE NATURE OF GEOMETRY
;277
22.1;§ 77. THE PROBLEM OF PARALLELISM. ELEMENTS AT INFINITY;278
22.2;§ 78. HOMOGENEOUS COORDINATES. CIRCULAR POINTS AT INFINITY
;280
22.3;§ 79. EUCLIDEAN GROUP. PROJECTIVE GEOMETRY
;285
22.4;§ 80. CROSS-RATIO
;289
22.5;EXERCISES 15;295
23;CHAPTER 16. CONICS
;298
23.1;§ 81. CONICS AS PLANE LOCI AND AS CONIC SECTIONS
;298
23.2;§ 82. GENERAL EQUATION OF A CONIC
;303
23.3;§
83. STANDARD EQUATIONS OF THE CONICS ;305
23.4;§
84. CONICS AND THE LINE AT INFINITY;307
23.5;§ 85. QUADRATIC EQUATION REPRESENTING A LINE-PAIR
;308
23.6;§ 8 6. TANGENT AT A GIVEN POINT;310
23.7;§ 87. ELEMENTARY THEORY OF POLE AND POLAR;310
23.8;§ 88. REDUCTION OF A CENTRAL CONIC TO STANDARD FORM
;313
23.9;EXERCISES 16;317
24;CHAPTER 17. THE PARABOLA
;319
24.1;§ 89. BASIC PROPERTIES (REVISION SUMMARY)
;319
24.2;§ 90. SELECTED PROBLEMS SOLVED PARAMETRICALLY
;321
24.3;§
91. NORMALS TO A PARABOLA;325
24.4;EXERCISES 17;328
25;CHAPTER 18. THE ELLIPSE
;330
25.1;§92. BASIC PROPERTIES;330
25.2;§ 93. SELECTED PROBLEMS SOLVED PARAMETRICALLY
;332
25.3;§ 94. CONJUGATE DIAMETERS
;334
25.4;EXERCISES 18;337
26;CHAPTER 19. THE HYPERBOLA
;340
26.1;§ 95. BASIC PROPERTIES
;340
26.2;§ 96. ASYMPTOTES
;342
26.3;§ 97. RECTANGULAR HYPERBOLA;344
26.4;EXERCISES 19;347
27;CHAPTER 20. CURVES: CARTESIAN COORDINATES
;350
27.1;§ 98. CONCAVITY, CONVEXITY, POINT OF INFLEXION
;350
27.2;§ 99. SOME RULES FOR CURVE-SKETCHING;351
27.3;§ 100. THE PROBLEM OF ASYMPTOTES
;353
27.4;§
101. DOUBLE POINTS;355
27.5;§ 102. SELECTED EXAMPLES OF CURVE-SKETCHING IN CARTESIAN COORDINATES
;355
27.6;§
103. COMPOSITION OF CURVES;358
27.7;§ 104. FAMILIES OF CURVES;359
27.8;§ 105. SPECIAL HIGHER PLANE CURVES
;360
27.9;§
106. PARAMETRIC CURVES;361
27.10;EXERCISES 20;368
28;CHAPTER 21. CURVATURE
;372
28.1;§ 107. INTRINSIC COORDINATES;372
28.2;§ 108. CURVATURE;373
28.3;§ 109. RADIUS OF CURVATURE;375
28.4;EXERCISES 21;378
29;CHAPTER 22. CURVES: POLAR COORDINATES
;380
29.1;§
110. EQUATIONS OF LINE AND CIRCLE IN POLARCOORDINATES;380
29.2;§ 111. EQUATIONS OF CONICS IN POLAR COORDINATES;381
29.3;§ 112. SOME RULES FOR CURVE-SKETCHING;383
29.4;§ 113. SELECTED EXAMPLES OF CURVE-SKETCHING IN POLAR COORDINATES;384
29.5;§ 114. SPIRALS AND ROSE CURVES;387
29.6;§ 115. MEANING OF rdq/dr
;391
29.7;§ 116. TANGENT IN POLAR COORDINATES;393
29.8;§
117. EQUIANGULAR SPIRAL;394
29.9;EXERCISES 22;397
30;CHAPTER 23. GEOMETRICAL APPLICATIONS OF THE DEFINITE INTEGRAL
;400
30.1;§ 118. AREA IN POLAR COORDINATES;400
30.2;§ 119. VOLUME OF A SOLID OF REVOLUTION;406
30.3;§ 120. LENGTH OF A CURVE
;409
30.4;§ 121. SURFACE AREA OF A SOLID OF REVOLUTION;413
30.5;§ 122. APPROXIMATE (NUMERICAL) INTEGRATION. CONCLUDING REMARKS
;415
30.6;EXERCISES 23;417
31;NEW HORIZONS;421
32;SOLUTIONS TO EXERCISES;424
33;INDEX;588