Calculus (Book): Difference between revisions
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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 | | 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> 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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| 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 | | 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230 | ||
|- | |- | ||
| 6.6 || Logarithms referred to any positive base | | 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 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 | ||
|- | |- | ||
| 6.14 || The definition of | | 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244 | ||
|- | |- | ||
| 6.15 || The definition of | | 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 | | 6.16 || Differentiation and integration formulas involving exponentials || 245 | ||
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| 7.13 || Exercises || 295 | | 7.13 || Exercises || 295 | ||
|- | |- | ||
| 7.14 || The symbols | | 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296 | ||
|- | |- | ||
| 7.15 || Infinite limits || 298 | | 7.15 || Infinite limits || 298 | ||
|- | |- | ||
| 7.16 || The behavior of log | | 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 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 | | 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 | | 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324 | ||
|- | |- | ||
| 8.11 || Uniqueness theorem for the equation | | 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324 | ||
|- | |- | ||
| 8.12 || Complete solution of the equation | | 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326 | ||
|- | |- | ||
| 8.13 || Complete solution of the equation | | 8.13 || Complete solution of the equation <math>y^{''} + ay^' + by = 0</math> || 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 | ||
|- | |- | ||
| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation | | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^' + by = R</math> || 332 | ||
|- | |- | ||
| 8.17 || Exercises || 333 | | 8.17 || 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 | | 9.4 || The imaginary unit <math>i</math> || 361 | ||
|- | |- | ||
| 9.5 || Geometric interpretation. Modulus and argument || 362 | | 9.5 || Geometric interpretation. Modulus and argument || 362 | ||
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| 12.2 || The vector space of n-tuples of real numbers || 446 | | 12.2 || The vector space of n-tuples of real numbers || 446 | ||
|- | |- | ||
| 12.3 || Geometric interpretation for | | 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448 | ||
|- | |- | ||
| 12.4 || Exercises || 450 | | 12.4 || Exercises || 450 | ||
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| 12.15 || Exercises || 467 | | 12.15 || Exercises || 467 | ||
|- | |- | ||
| 12.16 || The vector space | | 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468 | ||
|- | |- | ||
| 12.17 || Exercises || 470 | | 12.17 || Exercises || 470 |
Revision as of 17:04, 19 February 2023
Calculus | |
Information | |
---|---|
Author | Tom Apostol |
Language | English |
Publisher | Wiley |
Publication Date | 16 January 1991 |
Pages | 666 |
ISBN-10 | 0471000051 |
ISBN-13 | 978-0471000051 |
The textbook Calculus by 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 second volume, not listed here, which covers multivariable topics and applications to subjects such as probability.
Table of Contents
Chapter/Section # | Title | Page # |
---|---|---|
I. INTRODUCTION | ||
Part 1: Historical Introduction | ||
I 1.1 | The two basic concepts of calculus | 1 |
I 1.2 | Historical background | 2 |
I 1.3 | The method of exhaustion for the area of a parabolic segment | 3 |
*I 1.4 | Exercises | 8 |
I 1.5 | A critical analysis of the Archimedes' method | 8 |
I 1.6 | The approach to calculus to be used in this book | 10 |
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 |
Part 3: A set of Axioms for the Real-Number System | ||
I 3.1 | Introduction | 17 |
I 3.2 | The field axioms | 17 |
*I 3.3 | Exercises | 19 |
I 3.4 | The order axioms | 19 |
*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 |
*I 3.12 | Exercises | 28 |
*I 3.13 | Existence of square roots of nonnegative real numbers | 29 |
*I 3.14 | Roots of higher order. Rational powers | 30 |
*I 3.15 | Representation of real numbers by decimals | 30 |
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 |
*I 4.3 | The well-ordering principle | 34 |
I 4.4 | Exercises | 35 |
*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 |
*I 4.10 | Miscellaneous exercises involving induction | 44 |
1. THE CONCEPTS OF INTEGRAL CALCULUS | ||
1.1 | The basic ideas of Cartesian geometry | 48 |
1.2 | Functions. Informal description and examples | 50 |
*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 |
1.9 | Partitions and step functions | 61 |
1.10 | Sum and product of step functions | 63 |
1.11 | Exercises | 63 |
1.12 | The definition of the integral for step functions | 64 |
1.13 | Properties of the integral of a step function | 66 |
1.14 | Other notations for integrals | 69 |
1.15 | Exercises | 70 |
1.16 | The integral of more general functions | 72 |
1.17 | Upper and lower integrals | 74 |
1.18 | The area of an ordinate set expressed as an integral | 75 |
1.19 | Informal remarks on the theory and technique of integration | 75 |
1.20 | Monotonic and piecewise monotonic functions. Definitions and examples | 76 |
1.21 | Integrability of bounded monotonic functions | 77 |
1.22 | Calculation of the integral of a bounded monotonic function | 79 |
1.23 | Calculation of the integral [math]\displaystyle{ \int_0^b x^p dx }[/math] when [math]\displaystyle{ p }[/math] is a positive integer | 79 |
1.24 | The basic properties of the integral | 80 |
1.25 | Integration of polynomials | 81 |
1.26 | Exercises | 83 |
1.27 | Proofs of the basic properties of the integral | 84 |
2. SOME APPLICATIONS OF INTEGRATION | ||
2.1 | Introduction | 88 |
2.2 | The area of a region between two graphs expressed as an integral | 88 |
2.3 | Worked examples | 89 |
2.4 | Exercises | 94 |
2.5 | The trigonometric functions | 94 |
2.6 | Integration formulas for the sine and cosine | 94 |
2.7 | A geometric description of the sine and cosine functions | 94 |
2.8 | Exercises | 94 |
2.9 | Polar coordinates | 94 |
2.10 | The integral for area in polar coordinates | 94 |
2.11 | Exercises | 94 |
2.12 | Application of integration to the calculation of volume | 94 |
2.13 | Exercises | 94 |
2.14 | Application of integration to the calculation of work | 94 |
2.15 | Exercises | 94 |
2.16 | Average value of a function | 94 |
2.17 | Exercises | 94 |
2.18 | The integral as a function of the upper limit. Indefinite integrals | 94 |
2.19 | Exercises | 94 |
3. CONTINUOUS FUNCTIONS | ||
3.1 | Informal description of continuity | 126 |
3.2 | The definition of the limit of a function | 127 |
3.3 | The definition of continuity of a function | 130 |
3.4 | The basic limit theorems. More examples of continuous functions | 131 |
3.5 | Proofs of the basic limit theorems | 135 |
3.6 | Exercises | 138 |
3.7 | Composite functions and continuity | 140 |
3.8 | Exercises | 142 |
3.9 | Bolzano's theorem for continuous functions | 142 |
3.10 | The intermediate-value theorem for continuous functions | 144 |
3.11 | Exercises | 145 |
3.12 | The process of inversion | 146 |
3.13 | Properties of functions preserved by inversion | 147 |
3.14 | Inverses of piecewise monotonic functions | 148 |
3.15 | Exercises | 149 |
3.16 | The extreme-value theorem for continuous functions | 150 |
3.17 | The small-span theorem for continuous functions (uniform continuity) | 152 |
3.18 | The integrability theorem for continuous functions | 152 |
3.19 | Mean-value theorems for integrals of continuous functions | 154 |
3.20 | Exercises | 155 |
4. DIFFERENTIAL CALCULUS | ||
4.1 | Historical introduction | 156 |
4.2 | A problem involving velocity | 157 |
4.3 | The derivative of a function | 159 |
4.4 | Examples of derivatives | 161 |
4.5 | The algebra of derivatives | 164 |
4.6 | Exercises | 167 |
4.7 | Geometric interpretation of the derivative as a slope | 169 |
4.8 | Other notations for derivatives | 171 |
4.9 | Exercises | 173 |
4.10 | The chain rule for differentiating composite functions | 174 |
4.11 | Applications of the chain rule. Related rates and implicit differentiation | 176 |
4.12 | Exercises | 179 |
4.13 | Applications of the differentiation to extreme values of cuntions | 181 |
4.14 | The mean-value theorem for derivatives | 183 |
4.15 | Exercises | 186 |
4.16 | Applications of the mean-value theorem to geometric properties of functions | 187 |
4.17 | Second-derivative test for extrema | 188 |
4.18 | Curve sketching | 189 |
4.19 | Exercises | 191 |
4.20 | Worked examples of extremum problems | 191 |
4.21 | Exercises | 194 |
*4.22 | Partial derivatives | 196 |
*4.23 | Exercises | 201 |
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 |
*5.11 | Miscellaneous review exercises | 222 |
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]\displaystyle{ L(ab) = L(a) + L(b) }[/math] | 230 |
6.6 | Logarithms referred to any positive base [math]\displaystyle{ 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]\displaystyle{ e^x }[/math] for arbitrary real x | 244 |
6.15 | The definition of [math]\displaystyle{ a^x }[/math] for [math]\displaystyle{ a \gt 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 |
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 |
*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]\displaystyle{ +\inf }[/math] and [math]\displaystyle{ -\inf }[/math]. Extension of L'Hopital's rule | 296 |
7.15 | Infinite limits | 298 |
7.16 | The behavior of log[math]\displaystyle{ x }[/math] and [math]\displaystyle{ e^x }[/math] for large [math]\displaystyle{ x }[/math] | 300 |
7.17 | Exercises | 303 |
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]\displaystyle{ y^{''} + by = 0 }[/math] | 323 |
8.10 | Reduction of the general equation to the special case [math]\displaystyle{ y^{''} + by = 0 }[/math] | 324 |
8.11 | Uniqueness theorem for the equation [math]\displaystyle{ y^{''} + by = 0 }[/math] | 324 |
8.12 | Complete solution of the equation [math]\displaystyle{ y^{''} + by = 0 }[/math] | 326 |
8.13 | Complete solution of the equation [math]\displaystyle{ 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]\displaystyle{ 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 |
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]\displaystyle{ 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 |
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 |
*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 |
*10.21 | Rearrangements of series | 411 |
10.22 | Miscellaneous review exercises | 414 |
10.23 | Improper integrals | 416 |
10.24 | Exercises | 420 |
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 |
*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 |
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]\displaystyle{ 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]\displaystyle{ V_N(C) }[/math] of n-tuples of complex numbers | 468 |
12.17 | Exercises | 470 |
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 |
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 |
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 |
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 |
Answers to exercises | 617 | |
Index | 657 |