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| | {{Stub}} |
| {{InfoboxBook | | {{InfoboxBook |
| |title=Calculus | | |title=Calculus |
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| |isbn13=978-0471000051 | | |isbn13=978-0471000051 |
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| | The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. Β |
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| 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. | |
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| == Table of Contents == | | == Table of Contents == |
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| | 1.2 || Functions. Informal description and examples || 50 | | | 1.2 || Functions. Informal description and examples || 50 |
| |- | | |- |
| | <nowiki>*</nowiki>1.3 || Functions. Formal definition as a set of ordered pairs || 53 | | | 1.3 || Functions. Formal definition as a set of ordered pairs || 53 |
| |- | | |- |
| | 1.4 || More examples of real functions || 54 | | | 1.4 || More examples of real functions || 54 |
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| | 1.22 || Calculation of the integral of a bounded monotonic function || 79 | | | 1.22 || Calculation of the integral of a bounded monotonic function || 79 |
| |- | | |- |
| | 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79 | | | 1.23 || Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer || 79 |
| |- | | |- |
| | 1.24 || The basic properties of the integral || 80 | | | 1.24 || The basic properties of the integral || 80 |
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| | 4.21 || Exercises || 194 | | | 4.21 || Exercises || 194 |
| |- | | |- |
| | <nowiki>*</nowiki>4.22 || Partial derivatives || 196 | | | 4.22 || Partial derivatives || 196 |
| |- | | |- |
| | <nowiki>*</nowiki>4.23 || Exercises || 201 | | | 4.23 || Exercises || 201 |
| |- | | |- |
| ! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION | | ! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION |
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| | 5.10 || Exercises || 220 | | | 5.10 || Exercises || 220 |
| |- | | |- |
| | <nowiki>*</nowiki>5.11 || Miscellaneous review exercises || 222 | | | 5.11 || Miscellaneous review exercises || 222 |
| |- | | |- |
| ! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS | | ! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS |
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| | 6.4 || The graph of the natural logarithm || 230 | | | 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 | | | 6.5 || Consequences of the functional equation \(L(ab) = L(a) + L(b)\) || 230 |
| |- | | |- |
| | 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232 | | | 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 232 |
| |- | | |- |
| | 6.7 || Differentiation and integration formulas involving logarithms || 233 | | | 6.7 || Differentiation and integration formulas involving logarithms || 233 |
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| | 6.13 || Exponentials expressed as powers of e || 242 | | | 6.13 || Exponentials expressed as powers of e || 242 |
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| | 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244 | | | 6.14 || The definition of \(e^x\) for arbitrary real x || 244 |
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| | 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245 | | | 6.15 || The definition of \(a^x\) for \(a > 0\) and x real || 245 |
| |- | | |- |
| | 6.16 || Differentiation and integration formulas involving exponentials || 245 | | | 6.16 || Differentiation and integration formulas involving exponentials || 245 |
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| | 7.6 || Estimates for the error in Taylor's formula || 280 | | | 7.6 || Estimates for the error in Taylor's formula || 280 |
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| | <nowiki>*</nowiki>7.7 || Other forms of the remainder in Taylor's formula || 283 | | | 7.7 || Other forms of the remainder in Taylor's formula || 283 |
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| | 7.8 || Exercises || 284 | | | 7.8 || Exercises || 284 |
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| | 7.13 || Exercises || 295 | | | 7.13 || Exercises || 295 |
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| | 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296 | | | 7.14 || The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule || 296 |
| |- | | |- |
| | 7.15 || Infinite limits || 298 | | | 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 | | | 7.16 || The behavior of log\(x\) and \(e^x\) for large \(x\) || 300 |
| |- | | |- |
| | 7.17 || Exercises || 303 | | | 7.17 || Exercises || 303 |
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| | 8.8 || Linear equations of second order with constant coefficients || 322 | | | 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 | | | 8.9 || Existence of solutions of the equation \(y^{''} + by = 0\) || 323 |
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| | 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324 | | | 8.10 || Reduction of the general equation to the special case \(y^{''} + by = 0\) || 324 |
| |- | | |- |
| | 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324 | | | 8.11 || Uniqueness theorem for the equation \(y^{''} + by = 0\) || 324 |
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| | 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326 | | | 8.12 || Complete solution of the equation \(y^{''} + by = 0\) || 326 |
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| | 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326 | | | 8.13 || Complete solution of the equation \(y^{''} + ay^' + by = 0\) || 326 |
| |- | | |- |
| | 8.14 || Exercises || 328 | | | 8.14 || Exercises || 328 |
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| | 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 | | | 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 |
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| | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332 | | | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation \(y^{''} + ay^' + by = R\) || 332 |
| |- | | |- |
| | 8.17 || Exercises || 333 | | | 8.17 || Exercises || 333 |
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| | 9.3 || The complex numbers as an extension of the real numbers || 360 | | | 9.3 || The complex numbers as an extension of the real numbers || 360 |
| |- | | |- |
| | 9.4 || The imaginary unit <math>i</math> || 361 | | | 9.4 || The imaginary unit \(i\) || 361 |
| |- | | |- |
| | 9.5 || Geometric interpretation. Modulus and argument || 362 | | | 9.5 || Geometric interpretation. Modulus and argument || 362 |
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| | 9.10 || Exercises || 371 | | | 9.10 || Exercises || 371 |
| |- | | |- |
| ! colspan="3" | 10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS | | ! colspan="3" | PART IV: MISCELLANEOUS |
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| | 10.1 || Zeno's paradox || 374
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| | 10.2 || Sequences || 378
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| | 10.3 || Monotonic sequences of real numbers || 381
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| | 10.4 || Exercises || 382
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| | 10.5 || Infinite series || 383
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| | 10.6 || The linearity property of convergent series || 385
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| | 10.7 || Telescoping series || 386
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| | 10.8 || The geometric series || 388
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| | 10.9 || Exercises || 391
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| | <nowiki>*</nowiki>10.10 || Exercises on decimal expansions || 393
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| | 10.11 || Tests for convergence || 394
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| | 10.12 || Comparison tests for series of nonnegative terms || 394
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| | 10.13 || The integral test || 397
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| | 10.14 || Exercises || 398
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| | 10.15 || The root test and the ratio test for series of nonnegative terms || 399
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| | 10.16 || Exercises || 402
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| | 10.17 || Alternating series || 403
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| | 10.18 || Conditional and absolute convergence || 406
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| | 10.19 || The convergence tests of Dirichlet and Abel || 407
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| | 10.20 || Exercises || 409
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| | <nowiki>*</nowiki>10.21 || Rearrangements of series || 411
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| | 10.22 || Miscellaneous review exercises || 414
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| | 10.23 || Improper integrals || 416
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| | 10.24 || Exercises || 420
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| ! colspan="3" | 11. SEQUENCES AND SERIES OF FUNCTIONS
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| | 11.1 || Pointwise convergence of sequences of functions || 422
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| | 11.2 || Uniform convergence of sequences of functions || 423
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| | 11.3 || Uniform convergence and continuity || 424
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| | 11.4 || Uniform convergence and integration || 425
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| | 11.5 || A sufficient condition for uniform convergence || 427
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| | 11.6 || Power series. Circle of convergence || 428
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| | 11.7 || Exercises || 430
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| | 11.8 || Properties of functions represented by real power series || 431
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| | 11.9 || The Taylor's series generated by a function || 434
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| | 11.10 || A sufficient condition for convergence of a Taylor's series || 435
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| | 11.11 || Power-series expansions for the exponential and trigonometric functions || 435
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| | <nowiki>*</nowiki>11.12 || Bernstein's theorem || 437
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| | 11.13 || Exercises || 438
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| | 11.14 || Power series and differential equations || 439
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| | 11.15 || The binomial series || 441
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| | 11.16 || Exercises || 443
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| ! colspan="3" | 12. VECTOR ALGEBRA
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| | 12.1 || Historical introduction || 445
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| | 12.2 || The vector space of n-tuples of real numbers || 446
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| | 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448
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| | 12.4 || Exercises || 450
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| | 12.5 || The dot product || 451
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| | 12.6 || Length or norm of a vector|| 453
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| | 12.7 || Orthogonality of vectors || 455
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| | 12.8 || Exercises || 456
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| | 12.9 || Projections. Angle between vectors in n-space || 457
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| | 12.10 || The unit coordinate vectors || 458
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| | 12.11 || Exercises || 460
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| | 12.12 || The linear span of a finite set of vectors || 462
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| | 12.13 || Linear independence || 463
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| | 12.14 || Bases || 466
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| | 12.15 || Exercises || 467
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| | 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468
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| | 12.17 || Exercises || 470
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| ! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
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| |- | | |- |
| | 13.1 || Introduction || 471 | | ! colspan="3" | Chapter 13: Functions |
| |- | | |- |
| | 13.2 || Lines in n-space || 472 | | | 1 || Definition of a function || 313 |
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| | 13.3 || Some simple properties of straight lines || 473 | | | 2 || Polynomial functions || 318 |
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| | 13.4 || Lines and vector-valued functions || 474 | | | 3 || Graphs of functions || 330 |
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| | 13.5 || Exercises || 477 | | | 4 || Exponential function || 333 |
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| | 13.6 || Planes in Euclidean n-space || 478 | | | 5 || Logarithms || 338 |
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| | 13.7 || Planes and vector-valued functions || 481 | | ! colspan="3" | Chapter 14: Mappings |
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| | 13.8 || Exercises || 482 | | | 1 || Definition || 345 |
| |- | | |- |
| | 13.9 || The cross product || 483 | | | 2 || Formalism of mappings || 351 |
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| | 13.10 || The cross product expressed as a determinant || 486 | | | 3 || Permutations || 359 |
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| | 13.11 || Exercises || 487 | | ! colspan="3" | Chapter 15: Complex Numbers |
| |- | | |- |
| | 13.12 || The scalar triple product || 488 | | | 1 || The complex plane || 375 |
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| | 13.13 || Cramer's rule for solving a system of three linear equations || 490 | | | 2 || Polar form || 380 |
| |- | | |- |
| | 13.14 || Exercises || 491 | | ! colspan="3" | Chapter 16: Induction and Summations |
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| | 13.15 || Normal vectors to planes || 493 | | | 1 || Induction || 383 |
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| | 13.16 || Linear Cartesian equations for planes || 494 | | | 2 || Summations || 388 |
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| | 13.17 || Exercises || 496 | | | 3 || Geometric series || 396 |
| |- | | |- |
| | 13.18 || The conic sections || 497 | | ! colspan="3" | Chapter 17: Determinants |
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| | 13.19 || Eccentricity of conic sections || 500 | | | 1 || Matrices || 401 |
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| | 13.20 || Polar equations for conic sections || 501 | | | 2 || Determinants of order 2 || 406 |
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| | 13.21 || Exercises || 503 | | | 3 || Properties of 2 x 2 determinants || 409 |
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| | 13.22 || Conic sections symmetric about the origin || 504 | | | 4 || Determinants of order 3 || 414 |
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| | 13.23 || Cartesian equations for the conic sections || 505 | | | 5 || Properties of 3 x 3 determinants || 418 |
| |- | | |- |
| | 13.24 || Exercises || 508 | | | 6 || Cramer's Rule || 424 |
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| | 13.25 || Miscellaneous exercises on conic sections || 509
| | ! colspan="2" | Index || 429 |
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| ! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS
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| | 14.1 || Vector-valued functions of a real variable || 512
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| | 14.2 || Algebraic operations. Components || 512
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| | 14.3 || Limits, derivatives, and integrals || 513
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| | 14.4 || Exercises || 516
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| | 14.5 || Applications to curves. Tangency || 517
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| | 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520
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| | 14.7 || Exercises || 524
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| | 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525
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| | 14.9 || Exercises || 528
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| | 14.10 || The definition of arc length || 529
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| | 14.11 || Additivity of arc length || 532
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| | 14.12 || The arc-length function || 533
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| | 14.13 || Exercises || 535
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| | 14.14 || Curvature of a curve || 536
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| | 14.15 || Exercises || 538
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| | 14.16 || Velocity and acceleration in polar coordinates || 540
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| | 14.17 || Plane motion with radial acceleration || 542
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| | 14.18 || Cylindrical coordinates || 543
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| | 14.19 || Exercises || 543
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| | 14.20 || Applications to planetary motion || 545
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| | 14.21 || Miscellaneous review exercises || 549
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| ! colspan="3" | 15. LINEAR SPACES
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| | 15.1 || Introduction || 551
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| | 15.2 || The definition of a linear space || 551
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| | 15.3 || Examples of linear spaces || 552
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| | 15.4 || Elementary consequences of the axioms || 554
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| | 15.5 || Exercises || 555
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| | 15.6 || Subspaces of a linear space || 556
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| | 15.7 || Dependent and independent sets in a linear space || 557
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| | 15.8 || Bases and dimension || 559
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| | 15.9 || Exercises || 560
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| | 15.10 || Inner products, Euclidean spaces, norms || 561
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| | 15.11 || Orthogonality in a Euclidean space || 564
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| | 15.12 || Exercises || 566
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| | 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568
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| | 15.14 || Orthogonal complements. Projections || 572
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| | 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574
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| | 15.16 || Exercises || 576
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| ! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES
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| | 16.1 || Linear transformations || 578
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| | 16.2 || Null space and range || 579
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| | 16.3 || Nullity and rank || 581
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| | 16.4 || Exercises || 582
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| | 16.5 || Algebraic operations on linear transformations || 583
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| | 16.6 || Inverses || 585
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| | 16.7 || One-to-one linear transformations || 587
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| | 16.8 || Exercises || 589
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| | 16.9 || Linear transformations with prescribed values || 590
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| | 16.10 || Matrix representations of linear transformations || 591
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| | 16.11 || Construction of a matrix representation in diagonal form || 594
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| | 16.12 || Exercises || 596
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| | 16.13 || Linear spaces of matrices || 597
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| | 16.14 || Isomorphism between linear transformations and matrices || 599
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| | 16.15 || Multiplication of matrices || 600
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| | 16.16 || Exercises || 603
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| | 16.17 || Systems of linear equations || 605
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| | 16.18 || Computation techniques || 607
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| | 16.19 || Inverses of square matrices || 611
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| | 16.20 || Exercises || 613
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| | 16.21 || Miscellaneous exercises on matrices || 614
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| ! colspan="2" | Answers to exercises || 617
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| ! colspan="2" | Index || 657 | |
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| |} | | |} |
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| [[Category:Mathematics]] | | [[Category:Mathematics]] |
| {{Stub}}
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