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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Basic Mathematics
|title=Calculus
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|image=[[File:Apostol Calculus V1 Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|author=[https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol]
|language=English
|language=English
|series=
|series=
|genre=
|genre=
|publisher=Springer
|publisher=Wiley
|publicationdate=1 July 1988
|publicationdate=16 January 1991
|pages=496
|pages=666
|isbn10=0387967877
|isbn10=0471000051
|isbn13=978-0387967875
|isbn13=978-0471000051
}}
{{NavContainerFlex
|content=
{{NavButton|link=[[Read#Basic_Mathematics|Read]]}}
}}
}}
The textbook '''''Basic Mathematics''''' by [https://en.wikipedia.org/wiki/Serge_Lang Serge Lang] provides an overview of mathematical topics usually encountered through the end of high school/secondary school, specifically arithmetic, algebra, trigonometry, logic, and geometry. It serves as a solid review no matter how far along one may be in their studies, be it just beginning or returning to strengthen one's foundations.


Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.
The textbook [https://simeioseismathimatikwn.files.wordpress.com/2013/03/apostol-calculusi.pdf '''''Calculus'''''] by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. It provides a rigorous treatment of theory and application, in addition to the historical context of its topics. It should be noted that there is a [https://archive.org/details/calculus-tom-m.-apostol-calculus-volume-2-2nd-edition-proper-2-1975-wiley-sons-libgen.lc/Apostol%20T.%20M.%20-%20Calculus%20vol%20II%20%281967%29/ second volume], not listed here, which covers multivariable topics and applications to subjects such as probability.


== Table of Contents ==
== Table of Contents ==
Line 23: Line 25:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|-  
|-  
! colspan="3" | PART I: ALGEBRA
! colspan="3" | I. INTRODUCTION
|-
|-
! colspan="3" | Chapter 1: Numbers
! colspan="3" | Part 1: Historical Introduction
|-
|-
| 1 || The integers || 5
| I 1.1 || The two basic concepts of calculus || 1
|-
|-
| 2 || Rules for addition || 8
| I 1.2 || Historical background || 2
|-
|-
| 3 || Rules for multiplication || 14
| I 1.3 || The method of exhaustion for the area of a parabolic segment || 3
|-
|-
| 4 || Even and odd integers; divisibility || 22
| <nowiki>*</nowiki>I 1.4 || Exercises || 8
|-
|-
| 5 || Rational numbers || 26
| I 1.5 || A critical analysis of the Archimedes' method || 8
|-
|-
| 6 || Multiplicative inverses || 42
| I 1.6 || The approach to calculus to be used in this book || 10
|-  
|-  
! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | Part 2: Some Basic Concepts of the Theory of Sets
|-
| I 2.1 || Introduction to set theory || 11
|-
| I 2.2 || Notations for designating sets || 12
|-
| I 2.3 || Subsets || 12
|-
| I 2.4 || Unions, intersections, complements || 13
|-
| I 2.5 || Exercises || 15
|-
! colspan="3" | Part 3: A set of Axioms for the Real-Number System
|-
| I 3.1 || Introduction || 17
|-
| I 3.2 || The field axioms || 17
|-
| <nowiki>*</nowiki>I 3.3 || Exercises || 19
|-
| I 3.4 || The order axioms || 19
|-
| <nowiki>*</nowiki>I 3.5 || Exercises || 21
|-
| I 3.6 || Integers and rational numbers || 21
|-
| I 3.7 || Geometric interpretation of real numbers as points on a line || 22
|-
| I 3.8 || Upper bound of a set, maximum element, least upper bound (supremum) || 23
|-
| I 3.9 || The least-Upper-bound axiom (completeness axiom) || 25
|-
| I 3.10 || The Archimedean property of the real-number system || 25
|-
| I 3.11 || Fundamental properties of the supremum and infimum || 26
|-
| <nowiki>*</nowiki>I 3.12 || Exercises || 28
|-
| <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
|-
| <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
|-
| <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
|-
! colspan="3" | Part 4: Mathematical Induction, Summation Notation, and Related Topics
|-
| I 4.1 || An example of a proof by mathematical induction || 32
|-
| I 4.2 || The principle of mathematical induction || 34
|-
| <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
|-
| I 4.4 || Exercises || 35
|-
| <nowiki>*</nowiki>I 4.5 || Proof of the well-ordering principle || 37
|-
| I 4.6 || The summation notation || 37
|-
| I 4.7 || Exercises || 39
|-
| I 4.8 || Absolute values and the triangle inequality || 41
|-
| I 4.9 || Exercises || 43
|-
| <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
|-
! colspan="3" | 1. THE CONCEPTS OF INTEGRAL CALCULUS
|-
| 1.1 || The basic ideas of Cartesian geometry || 48
|-
| 1.2 || Functions. Informal description and examples || 50
|-
| <nowiki>*</nowiki>1.3 || Functions. Formal definition as a set of ordered pairs || 53
|-
| 1.4 || More examples of real functions || 54
|-
| 1.5 || Exercises || 56
|-
| 1.6 || The concept of area as a set function || 57
|-
| 1.7 || Exercises || 60
|-
| 1.8 || Intervals and ordinate sets || 60
|-
|-
| 1 || Equations in two unknowns || 53
| 1.9 || Partitions and step functions || 61
|-
|-
| 2 || Equations in three unknowns || 57
| 1.10 || Sum and product of step functions || 63
|-
|-
! colspan="3" | Chapter 3: Real Numbers
| 1.11 || Exercises || 63
|-
|-
| 1 || Addition and multiplication || 61
| 1.12 || The definition of the integral for step functions || 64
|-
|-
| 2 || Real numbers: positivity || 64
| 1.13 || Properties of the integral of a step function || 66
|-
|-
| 3 || Powers and roots || 70
| 1.14 || Other notations for integrals || 69
|-
|-
| 4 || Inequalities || 75
| 1.15 || Exercises || 70
|-
|-
! colspan="3" | Chapter 4: Quadratic Equations
| 1.16 || The integral of more general functions || 72
|-
|-
! colspan="3" | Interlude: On Logic and Mathematical Expressions
| 1.17 || Upper and lower integrals || 74
|-
|-
| 1 || On reading books || 93
| 1.18 || The area of an ordinate set expressed as an integral || 75
|-
|-
| 2 || Logic || 94
| 1.19 || Informal remarks on the theory and technique of integration || 75
|-
|-
| 3 || Sets and elements || 99
| 1.20 || Monotonic and piecewise monotonic functions. Definitions and examples || 76
|-
|-
| 4 || Notation || 100
| 1.21 || Integrability of bounded monotonic functions || 77
|-
|-
! colspan="3" | PART II: INTUITIVE GEOMETRY
| 1.22 || Calculation of the integral of a bounded monotonic function || 79
|-
|-
! colspan="3" | Chapter 5: Distance and Angles
| 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79
|-
|-
| 1 || Distance || 107
| 1.24 || The basic properties of the integral || 80
|-
|-
| 2 || Angles || 110
| 1.25 || Integration of polynomials || 81
|-
|-
| 3 || The Pythagoras theorem || 120
| 1.26 || Exercises || 83
|-
|-
! colspan="3" | Chapter 6: Isometries
| 1.27 || Proofs of the basic properties of the integral || 84
|-
|-
| 1 || Some standard mappings of the plane || 133
! colspan="3" | 2. SOME APPLICATIONS OF INTEGRATION
|-
|-
| 2 || Isometries || 143
| 2.1 || Introduction || 88
|-
|-
| 3 || Composition of isometries || 150
| 2.2 || The area of a region between two graphs expressed as an integral || 88
|-
|-
| 4 || Inverse of isometries || 155
| 2.3 || Worked examples || 89
|-
|-
| 5 || Characterization of isometries || 163
| 2.4 || Exercises || 94
|-
|-
| 6 || Congruences || 166
| 2.5 || The trigonometric functions || 94
|-
|-
! colspan="3" | Chapter 7: Area and Applications
| 2.6 || Integration formulas for the sine and cosine || 94
|-
|-
| 1 || Area of a disc of radius ''r'' || 173
| 2.7 || A geometric description of the sine and cosine functions || 94
|-
|-
| 2 || Circumference of a circle of radius ''r'' || 180
| 2.8 || Exercises || 94
|-
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
| 2.9 || Polar coordinates || 94
|-
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
| 2.10 || The integral for area in polar coordinates || 94
|-
|-
| 1 || Coordinate systems || 191
| 2.11 || Exercises || 94
|-
|-
| 2 || Distance between points || 197
| 2.12 || Application of integration to the calculation of volume || 94
|-
|-
| 3 || Equation of a circle || 203
| 2.13 || Exercises || 94
|-
|-
| 4 || Rational points on a circle || 206
| 2.14 || Application of integration to the calculation of work || 94
|-
|-
! colspan="3" | Chapter 9: Operations on Points
| 2.15 || Exercises || 94
|-
|-
| 1 || Dilations and reflections || 213
| 2.16 || Average value of a function || 94
|-
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
| 2.17 || Exercises || 94
|-
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
| 2.18 || The integral as a function of the upper limit. Indefinite integrals || 94
|-
|-
| 1 || Segments || 229
| 2.19 || Exercises || 94
|-
|-
| 2 || Rays || 231
! colspan="3" | 3. CONTINUOUS FUNCTIONS
|-
|-
| 3 || Lines || 236
| 3.1 || Informal description of continuity || 126
|-
|-
| 4 || Ordinary equation for a line || 246
| 3.2 || The definition of the limit of a function || 127
|-
|-
! colspan="3" | Chapter 11: Trigonometry
| 3.3 || The definition of continuity of a function || 130
|-
|-
| 1 || Radian measure || 249
| 3.4 || The basic limit theorems. More examples of continuous functions || 131
|-
|-
| 2 || Sine and cosine || 252
| 3.5 || Proofs of the basic limit theorems || 135
|-
|-
| 3 || The graphs || 264
| 3.6 || Exercises || 138
|-
|-
| 4 || The tangent || 266
| 3.7 || Composite functions and continuity || 140
|-
|-
| 5 || Addition formulas || 272
| 3.8 || Exercises || 142
|-
|-
| 6 || Rotations || 277
| 3.9 || Bolzano's theorem for continuous functions || 142
|-
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
| 3.10 || The intermediate-value theorem for continuous functions || 144
|-
|-
| 1 || The straight line again || 281
| 3.11 || Exercises || 145
|-
|-
| 2 || The parabola || 291
| 3.12 || The process of inversion || 146
|-
|-
| 3 || The ellipse || 297
| 3.13 || Properties of functions preserved by inversion || 147
|-
|-
| 4 || The hyperbola || 300
| 3.14 || Inverses of piecewise monotonic functions || 148
|-
|-
| 5 || Rotation of hyperbolas || 305
| 3.15 || Exercises || 149
|-
|-
! colspan="3" | PART IV: MISCELLANEOUS
| 3.16 || The extreme-value theorem for continuous functions || 150
|-
|-
! colspan="3" | Chapter 13: Functions
| 3.17 || The small-span theorem for continuous functions (uniform continuity) || 152
|-
|-
| 1 || Definition of a function || 313
| 3.18 || The integrability theorem for continuous functions || 152
|-
|-
| 2 || Polynomial functions || 318
| 3.19 || Mean-value theorems for integrals of continuous functions || 154
|-
|-
| 3 || Graphs of functions || 330
| 3.20 || Exercises || 155
|-
|-
| 4 || Exponential function || 333
! colspan="3" | 4. DIFFERENTIAL CALCULUS
|-
|-
| 5 || Logarithms || 338
| 4.1 || Historical introduction || 156
|-
|-
! colspan="3" | Chapter 14: Mappings
| 4.2 || A problem involving velocity || 157
|-
|-
| 1 || Definition || 345
| 4.3 || The derivative of a function || 159
|-
|-
| 2 || Formalism of mappings || 351
| 4.4 || Examples of derivatives || 161
|-
|-
| 3 || Permutations || 359
| 4.5 || The algebra of derivatives || 164
|-
|-
! colspan="3" | Chapter 15: Complex Numbers
| 4.6 || Exercises || 167
|-
|-
| 1 || The complex plane || 375
| 4.7 || Geometric interpretation of the derivative as a slope || 169
|-
|-
| 2 || Polar form || 380
| 4.8 || Other notations for derivatives || 171
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
| 4.9 || Exercises || 173
|-
|-
| 1 || Induction || 383
| 4.10 || The chain rule for differentiating composite functions || 174
|-
|-
| 2 || Summations || 388
| 4.11 || Applications of the chain rule. Related rates and implicit differentiation || 176
|-
|-
| 3 || Geometric series || 396
| 4.12 || Exercises || 179
|-
|-
! colspan="3" | Chapter 17: Determinants
| 4.13 || Applications of the differentiation to extreme values of cuntions|| 181
|-
|-
| 1 || Matrices || 401
| 4.14 || The mean-value theorem for derivatives || 183
|-
|-
| 2 || Determinants of order 2 || 406
| 4.15 || Exercises || 186
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| 4.16 || Applications of the mean-value theorem to geometric properties of functions || 187
|-
|-
| 4 || Determinants of order 3 || 414
| 4.17 || Second-derivative test for extrema || 188
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
| 4.18 || Curve sketching || 189
|-
|-
| 6 || Cramer's Rule || 424
| 4.19 || Exercises || 191
|-
|-
! colspan="2" | Index || 429
| 4.20 || Worked examples of extremum problems || 191
|-
| 4.21 || Exercises || 194
|-
| <nowiki>*</nowiki>4.22 || Partial derivatives || 196
|-
| <nowiki>*</nowiki>4.23 || Exercises || 201
|-
! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
|-
| 5.1 || The derivative of an indefinite integral. The first fundamental theorem of calculus || 202
|-
| 5.2 || The zero-derivative theorem || 204
|-
| 5.3 || Primitive functions and the second fundamental theorem of calculus || 205
|-
| 5.4 || Properties of a function deduced from properties of its derivative || 207
|-
| 5.5 || Exercises || 208
|-
| 5.6 || The Leibniz notation for primitives || 210
|-
| 5.7 || Integration by substitution || 212
|-
| 5.8 || Exercises || 216
|-
| 5.9 || Integration by parts || 217
|-
| 5.10 || Exercises || 220
|-
| <nowiki>*</nowiki>5.11 || Miscellaneous review exercises || 222
|-
! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
|-
| 6.1 || Introduction || 226
|-
| 6.2 || Motivation for the definition of the natural logarithm as an integral || 227
|-
| 6.3 || The definition of the logarithm. Basic properties || 229
|-
| 6.4 || The graph of the natural logarithm || 230
|-
| 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230
|-
| 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232
|-
| 6.7 || Differentiation and integration formulas involving logarithms || 233
|-
| 6.8 || Logarithmic differentiation || 235
|-
| 6.9 || Exercises || 236
|-
| 6.10 || Polynomial approximations to the logarithm || 236
|-
| 6.11 || Exercises || 242
|-
| 6.12 || The exponential function || 242
|-
| 6.13 || Exponentials expressed as powers of e || 242
|-
| 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244
|-
| 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245
|-
| 6.16 || Differentiation and integration formulas involving exponentials || 245
|-
| 6.17 || Exercises || 248
|-
| 6.18 || The hyperbolic functions || 251
|-
| 6.19 || Exercises || 251
|-
| 6.20 || Derivatives of inverse functions || 252
|-
| 6.21 || Inverses of the trigonometric functions || 253
|-
| 6.22 || Exercises || 256
|-
| 6.23 || Integration by partial fractions || 258
|-
| 6.24 || Integrals which can be transformed into integrals of rational functions || 264
|-
| 6.25 || Exercises || 267
|-
| 6.26 || Miscellaneous review exercises || 268
|-
! colspan="3" | 7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
|-
| 7.1 || Introduction || 272
|-
| 7.2 || The Taylor polynomials generated by a function || 273
|-
| 7.3 || Calculus of Taylor polynomials || 275
|-
| 7.4 || Exercises || 278
|-
| 7.5 || Taylor's formula with remainder || 278
|-
| 7.6 || Estimates for the error in Taylor's formula || 280
|-
| <nowiki>*</nowiki>7.7 || Other forms of the remainder in Taylor's formula || 283
|-
| 7.8 || Exercises || 284
|-
| 7.9 || Further remarks on the error in Taylor's formula. The o-notation || 286
|-
| 7.10 || Applications to indeterminate forms || 289
|-
| 7.11 || Exercises || 290
|-
| 7.12 || L'Hopital's rule for the indeterminate form 0/0 || 292
|-
| 7.13 || Exercises || 295
|-
| 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296
|-
| 7.15 || Infinite limits || 298
|-
| 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300
|-
| 7.17 || Exercises || 303
|-
! colspan="3" | 8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
|-
| 8.1 || Introduction || 305
|-
| 8.2 || Terminology and notation || 306
|-
| 8.3 || A first-order differential equation for the exponential function || 307
|-
| 8.4 || First-order linear differential equations || 308
|-
| 8.5 || Exercises || 311
|-
| 8.6 || Some physical problems leading to first-order linear differential equations || 313
|-
| 8.7 || Exercises || 319
|-
| 8.8 || Linear equations of second order with constant coefficients || 322
|-
| 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323
|-
| 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324
|-
| 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324
|-
| 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326
|-
| 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326
|-
| 8.14 || Exercises || 328
|-
| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329
|-
| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332
|-
| 8.17 || Exercises || 333
|-
| 8.18 || Examples of physical problems leading to linear second-order equations with constant coefficients || 334
|-
| 8.19 || Exercises || 339
|-
| 8.20 || Remarks concerning nonlinear differential equations || 339
|-
| 8.21 || Integral curves and direction fields || 341
|-
| 8.22 || Exercises || 344
|-
| 8.23 || First-order separable equations || 345
|-
| 8.24 || Exercises || 347
|-
| 8.25 || Homogeneous first-order equations || 347
|-
| 8.26 || Exercises || 350
|-
| 8.27 || Some geometrical and physical problems leading to first-order equations || 351
|-
| 8.28 || Miscellaneous review exercises || 355
|-
! colspan="3" | 9. COMPLEX NUMBERS
|-
| 9.1 || Historical introduction || 358
|-
| 9.2 || Definitions and field properties || 358
|-
| 9.3 || The complex numbers as an extension of the real numbers || 360
|-
| 9.4 || The imaginary unit <math>i</math> || 361
|-
| 9.5 || Geometric interpretation. Modulus and argument || 362
|-
| 9.6 || Exercises || 365
|-
| 9.7 || Complex exponentials || 366
|-
| 9.8 || Complex-valued functions || 368
|-
| 9.9 || Examples of differentiation and integration formulas || 369
|-
| 9.10 || Exercises || 371
|-
! colspan="3" | 10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS
|-
| 10.1 || Zeno's paradox || 374
|-
| 10.2 || Sequences || 378
|-
| 10.3 || Monotonic sequences of real numbers || 381
|-
| 10.4 || Exercises || 382
|-
| 10.5 || Infinite series || 383
|-
| 10.6 || The linearity property of convergent series || 385
|-
| 10.7 || Telescoping series || 386
|-
| 10.8 || The geometric series || 388
|-
| 10.9 || Exercises || 391
|-
| <nowiki>*</nowiki>10.10 || Exercises on decimal expansions || 393
|-
| 10.11 || Tests for convergence || 394
|-
| 10.12 || Comparison tests for series of nonnegative terms || 394
|-
| 10.13 || The integral test || 397
|-
| 10.14 || Exercises || 398
|-
| 10.15 || The root test and the ratio test for series of nonnegative terms || 399
|-
| 10.16 || Exercises || 402
|-
| 10.17 || Alternating series || 403
|-
| 10.18 || Conditional and absolute convergence || 406
|-
| 10.19 || The convergence tests of Dirichlet and Abel || 407
|-
| 10.20 || Exercises || 409
|-
| <nowiki>*</nowiki>10.21 || Rearrangements of series || 411
|-
| 10.22 || Miscellaneous review exercises || 414
|-
| 10.23 || Improper integrals || 416
|-
| 10.24 || Exercises || 420
|-
! colspan="3" | 11. SEQUENCES AND SERIES OF FUNCTIONS
|-
| 11.1 || Pointwise convergence of sequences of functions || 422
|-
| 11.2 || Uniform convergence of sequences of functions || 423
|-
| 11.3 || Uniform convergence and continuity || 424
|-
| 11.4 || Uniform convergence and integration || 425
|-
| 11.5 || A sufficient condition for uniform convergence || 427
|-
| 11.6 || Power series. Circle of convergence || 428
|-
| 11.7 || Exercises || 430
|-
| 11.8 || Properties of functions represented by real power series || 431
|-
| 11.9 || The Taylor's series generated by a function || 434
|-
| 11.10 || A sufficient condition for convergence of a Taylor's series || 435
|-
| 11.11 || Power-series expansions for the exponential and trigonometric functions || 435
|-
| <nowiki>*</nowiki>11.12 || Bernstein's theorem || 437
|-
| 11.13 || Exercises || 438
|-
| 11.14 || Power series and differential equations || 439
|-
| 11.15 || The binomial series || 441
|-
| 11.16 || Exercises || 443
|-
! colspan="3" | 12. VECTOR ALGEBRA
|-
| 12.1 || Historical introduction || 445
|-
| 12.2 || The vector space of n-tuples of real numbers || 446
|-
| 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448
|-
| 12.4 || Exercises || 450
|-
| 12.5 || The dot product || 451
|-
| 12.6 || Length or norm of a vector|| 453
|-
| 12.7 || Orthogonality of vectors || 455
|-
| 12.8 || Exercises || 456
|-
| 12.9 || Projections. Angle between vectors in n-space || 457
|-
| 12.10 || The unit coordinate vectors || 458
|-
| 12.11 || Exercises || 460
|-
| 12.12 || The linear span of a finite set of vectors || 462
|-
| 12.13 || Linear independence || 463
|-
| 12.14 || Bases || 466
|-
| 12.15 || Exercises || 467
|-
| 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468
|-
| 12.17 || Exercises || 470
|-
! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
|-
| 13.1 || Introduction || 471
|-
| 13.2 || Lines in n-space || 472
|-
| 13.3 || Some simple properties of straight lines || 473
|-
| 13.4 || Lines and vector-valued functions || 474
|-
| 13.5 || Exercises || 477
|-
| 13.6 || Planes in Euclidean n-space || 478
|-
| 13.7 || Planes and vector-valued functions || 481
|-
| 13.8 || Exercises || 482
|-
| 13.9 || The cross product || 483
|-
| 13.10 || The cross product expressed as a determinant || 486
|-
| 13.11 || Exercises || 487
|-
| 13.12 || The scalar triple product || 488
|-
| 13.13 || Cramer's rule for solving a system of three linear equations || 490
|-
| 13.14 || Exercises || 491
|-
| 13.15 || Normal vectors to planes || 493
|-
| 13.16 || Linear Cartesian equations for planes || 494
|-
| 13.17 || Exercises || 496
|-
| 13.18 || The conic sections || 497
|-
| 13.19 || Eccentricity of conic sections || 500
|-
| 13.20 || Polar equations for conic sections || 501
|-
| 13.21 || Exercises || 503
|-
| 13.22 || Conic sections symmetric about the origin || 504
|-
| 13.23 || Cartesian equations for the conic sections || 505
|-
| 13.24 || Exercises || 508
|-
| 13.25 || Miscellaneous exercises on conic sections || 509
|-
! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS
|-
| 14.1 || Vector-valued functions of a real variable || 512
|-
| 14.2 || Algebraic operations. Components || 512
|-
| 14.3 || Limits, derivatives, and integrals || 513
|-
| 14.4 || Exercises || 516
|-
| 14.5 || Applications to curves. Tangency || 517
|-
| 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520
|-
| 14.7 || Exercises || 524
|-
| 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525
|-
| 14.9 || Exercises || 528
|-
| 14.10 || The definition of arc length || 529
|-
| 14.11 || Additivity of arc length || 532
|-
| 14.12 || The arc-length function || 533
|-
| 14.13 || Exercises || 535
|-
| 14.14 || Curvature of a curve || 536
|-
| 14.15 || Exercises || 538
|-
| 14.16 || Velocity and acceleration in polar coordinates || 540
|-
| 14.17 || Plane motion with radial acceleration || 542
|-
| 14.18 || Cylindrical coordinates || 543
|-
| 14.19 || Exercises || 543
|-
| 14.20 || Applications to planetary motion || 545
|-
| 14.21 || Miscellaneous review exercises || 549
|-
! colspan="3" | 15. LINEAR SPACES
|-
| 15.1 || Introduction || 551
|-
| 15.2 || The definition of a linear space || 551
|-
| 15.3 || Examples of linear spaces || 552
|-
| 15.4 || Elementary consequences of the axioms || 554
|-
| 15.5 || Exercises || 555
|-
| 15.6 || Subspaces of a linear space || 556
|-
| 15.7 || Dependent and independent sets in a linear space || 557
|-
| 15.8 || Bases and dimension || 559
|-
| 15.9 || Exercises || 560
|-
| 15.10 || Inner products, Euclidean spaces, norms || 561
|-
| 15.11 || Orthogonality in a Euclidean space || 564
|-
| 15.12 || Exercises || 566
|-
| 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568
|-
| 15.14 || Orthogonal complements. Projections || 572
|-
| 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574
|-
| 15.16 || Exercises || 576
|-
! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES
|-
| 16.1 || Linear transformations || 578
|-
| 16.2 || Null space and range || 579
|-
| 16.3 || Nullity and rank || 581
|-
| 16.4 || Exercises || 582
|-
| 16.5 || Algebraic operations on linear transformations || 583
|-
| 16.6 || Inverses || 585
|-
| 16.7 || One-to-one linear transformations || 587
|-
| 16.8 || Exercises || 589
|-
| 16.9 || Linear transformations with prescribed values || 590
|-
| 16.10 || Matrix representations of linear transformations || 591
|-
| 16.11 || Construction of a matrix representation in diagonal form || 594
|-
| 16.12 || Exercises || 596
|-
| 16.13 || Linear spaces of matrices || 597
|-
| 16.14 || Isomorphism between linear transformations and matrices || 599
|-
| 16.15 || Multiplication of matrices || 600
|-
| 16.16 || Exercises || 603
|-
| 16.17 || Systems of linear equations || 605
|-
| 16.18 || Computation techniques || 607
|-
| 16.19 || Inverses of square matrices || 611
|-
| 16.20 || Exercises || 613
|-
| 16.21 || Miscellaneous exercises on matrices || 614
|-
! colspan="2" | Answers to exercises || 617
|-
! colspan="2" | Index || 657
|-
|-
|}
|}


[[Category:Mathematics]]
[[Category:Mathematics]]
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