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© in This Web Service Cambridge University Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information INDEX © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information AalbmcndoE, 273 Archimedian property Abel, Niels Henrik, 404, 513 for the rational numbers, 574 Abel summable, 514 for the real numbers, 136 Abel's formula for summation by parts, Archimedian spiral, 85, 246 388 Arcsec, 317, 379 Abel's Lemma, 389 Arcsin, 307 Abel's test, 488 addition formula for, 314 Abel's Theorem, 513 derivative of, 308 Absolute value, 11 Taylor series for, 509 of a complex number, 525 Arctan, 308 Absolutely convergent, 473, 547 addition formula for, 314 Absolutely summable, 473 derivative of, 308 Acceleration, 159 Taylor polynomial for, 407, 414 remainder term for, 422 Acta Eruditorum, 146 Addition, 3 Area, 250, 255 associative law for, 9 Arg cosh, 350 commutative law for, 9 Arg sinh, 350 of complex numbers, 522 Arg tanh, 350 Argument, 527 geometric interpretation of, 526 Argument function, 533 of vector-valued functions, 242 discontinuities of, 537 of vectors, 75 Argument of the hyperbolic Addition formula functions, 350 for arcsin, 314 Arithmetic mean, 33 for arctan, 314 Arrow, 75, 76 for cos, 311 \x arrow sin(x2)", 45 for sin, 310, 311 Associative law for tan, 314 for addition, 9 Additive identity of vectors, 76 existence of, 9 for multiplication, 9 for vectors, 76 Average velocity, 150 Additive inverses Axis existence of, 9 horizontal, 57 Algebra, Fundamental Theorem of, imaginary, 525 373, 529, 539, 558 real, 524 Algebraic functions, 359 vertical, 57 Algebraic number, 435 Algebraist's real numbers, 588 Almost lower bound, 140 Almost upper bound, 140 Bacon, Francis, vi Analyst's real numbers, 588 Basic properties of numbers, 3 Angle, 300 \Bent graphs", 147 directed, 300 Bernoulli, 146, 565 Antidiagonal, 239 Bernoulli numbers, 563 Arabic numerals, multiplication of, 8 Bernoulli polynomials, 566 Arc length, 275, 281 Bernoulli's inequality, 32 Arccos, 308 Big game hunting, mathematical derivative of, 308 theory of, 543 Archimedes, 136, 139, 260 Binary operation, 571 659 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information 660 Index Binomial coefˇcient, 27, 429 Closed rectangle, 538 Binomial series, 487, 510 Closure under addition, 9 Binomial theorem, 28 Closure under multiplication, 9 Bisection argument, 140, 543 Commutative law Bohr, Harold, 390 for addition, 9 Bolzano-Weierstrass Theorem, 451, 461, of vectors, 76 543 for multiplication, 9 Bound Comparison test, 467, 468 almost lower, 140 Comparison Theorem, Sturm, 320 almost upper, 140 Complete induction, 23 greatest lower, 132 Complete ordered ˇeld, 574, 593 least upper, 131, 574 Completing the square, 17, 375 lower, 132 Complex analysis, 556 upper, 131 Complex function Bounded above, 120, 131, 450, 574 continuous, 536 Bounded below, 132, 450 differentiable, 541 Bourbaki, Nicholas, 146 graph of, 533 limit of, 533 nondifferentiable, 542 Taylor series for, 554 Cardioid, 89, 247 Complex nth root, 527 Cartesian coordinates, 84 Complex numbers, 517, 522 Cauchy, 278 absolute value of, 525 Cauchy Condensation Theorem, 488 addition of, 522 Cauchy criterion, 466 geometric interpretation of, 526 Cauchy form of the remainder, 417, geometric interpretation of, 525 419 imaginary part of, 522 Cauchy Mean Value Theorem, 201 inˇnite sequence of, 546 Cauchy product, 486, 505 inˇnite series of, 546{548 Cauchy sequence, 452, 562 logarithm of, 561 equivalence of, 589 modulus of, 525 Cauchy-Hadamard formula, 560 multiplication of, 522 Cauchy-Schwarz inequality, 278 geometric interpretation Cavalieri, 272 of, 526{527 Cesaro summable, 486 real part of, 522 Chain Rule, 172 ff. Complex plane, 524 proof of, 176 Complex power series, 548 Change, rate of, 150 circle of convergence of, 550 Characteristic (of a ˇeld), 576 radius of convergence of, 550 Circle, 65 Complex-valued functions, 532 \f circle g", 44 Composition of functions, 44 unit, 66 Concave function, 217 Circle of convergence, 550 Conditionally convergent series, 474 Classical notation Cone, 80 for derivatives, 152{154, 160, 165, generating line of, 80 184, 238 surface area of, 399 for integrals, 262 Conic sections, 80; see also Ellipse, Cleio, 183 Hyperbola, Parabola Closed interval, 57 Conjugate, 525, 530 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information Index 661 Conjugate function, 532 derivative of, 307 Constant function, 43 Cubic equation, general solution, Construction of the real numbers, 519{520 578 ff. Curve Continued fraction, 455 parameterized, tangent line of, 243 Continuous, uniformly, 142 parametric representation of, 241 Continuous at a, 113, 536 reparameterization of, 244 Continuous function, 113, 116, 537 Cycloid, 247 nowhere differentiable, 157, 501 Continuous on (a, b), 116 Continuous on [a,b], 116 Contraction, 459 Darboux's Theorem, 211 Contraction lemma, 459 De Moivre's Theorem, 527 Converge Decimal expansion, 73, 485 pointwise, 494 Decreasing function, 192 uniformly, 494, 498 Decreasing sequence, 450 Convergent sequence, 446, 546 Dedekind, Richard, 38 Convergent series, 465, 547 Deˇned implicitly, 238 absolutely, 473, 547 Deˇnite integral, 361 conditionally, 474 DEFINITION,47 Convex function, 216 Deˇnition, recursive, 23 strictly, 226 Degree (of a polynomial), 42 weakly, 226 Degree measurement, 63, 301{302 Convex subset of the plane, 226, 544 Delicate ratio test, 486 Cooling, Newton's law of, 352 Delicate root test, 486 Coordinate Dense, 138 ˇrst, 57 Derivative, 147 ff., 149 second, 57 classical notation for, 152{154, 160, Coordinate system, 57 165, 184, 238 cartesian, 84 higher-order, 159 origin of, 57 \inˇnite", 156 Coordinates left-hand, 154 polar, 84 Leibnizian notation for, see Derivative, \Corner", 60 classical notation for Cos, 300, 303, 318{319, 554 logarithmic, 348 addition formula for, 311 \negative inˇnity", 156 derivative of, 170, 304 of f , 149 inverse of, see Arccos of f at a, 149 Taylor polynomials for, 407 of vector-valued function, 243 remainder term for, 420 right-hand, 154 Cosh, 349 Schwarzian, 182 Cosine, hyperbolic, 349 second, 159 Cot, 307 Schwarz, 431 derivative of, 307 Derivative of quotient, incantation for, Countable, 442 169 Counting numbers, 21 Descartes, Rene,à 84 Critical point, 187 Determinant, 79 Critical value, 187 of vector-valued functions, 243 Csc, 307 Diagonal, 230 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information 662 Index Difference operator, 564 Entire function, 558 Differentiable, 149, 541 Epsilon, 18 Differential equation, 289, 297, 318, Equal up to order n, 412 320, 352, 357, 432 Equality, order of, 412 initial conditions for, 433 Equations, differential, see Differential Differentiation, 166 ff. equations implicit, 238 Equivalent Cauchy sequences, 589 logarithmic, 348 Etymology lesson, 82 Differentiation operator, 564 Euler, 565 Dini's Theorem, 515 Euler's number, 456 Directed angle, 300 Euler-Maclaurin Summation Formula, Dirichlet's test, 488 566 Disc method, 397 Even function, 51, 196 Discontinuities of a nondecreasing Even number, 25 function, 443 Eventually inside, 546 Discontinuity, removable, 119 Exhaustion, method of, 139 Disraeli, Benjamin, 2 Exp, 340 ff., 554 Distance, 58, 525 classical approach to, 354 shortest between two points, 275 elementary deˇnition of, 461 Distributive law, 9 Taylor polynomials for, 407 Diverge, 446, 547 remainder term for, 422 Division, 6 Expansion, decimal, 73, 485 Division by zero, 6 Extension of a function, 113{114 Domain, 40, 41, 47, 591 Dot product of vectors, 78 of vector-valued functions, 243 Double intersection, 163 Factorial, 23 Double root, 183 Factorials, table of, 428 Durege, 38 Factorization into primes, 31 Fibonacci, 32 Fibonacci Association, 32 Fibonacci Quarterly, The,32 e, 340 Fibonacci sequence, 32, 512, 563 irrationality of, 425 Field, 571 relation with π, 441, 555 characteristic of, 576 transcendentality of, 437 complete ordered, 574, 593 value of, 341, 422 ordered, 573 Eccentricity of ellipse, 87 First coordinate, 57 Elementary function, 359 First Fundamental Theorem of Calculus, Ellipse, 66, 82 282 axes of, 87 Fixed point of a function, 458 eccentricity of, 87 Focus point, 66, 86 equation in polar coordinates, 86 Force, as vector, 76 focus point of, 66, 86 Four leaf clover, 88 major axis of, 87 Fourier series, 315, 317, 320 minor axis of, 87 Fraction, continued, 455 Ellipsoid of revolution, 400 Function, 39, 47 Empty collection, 23 absolute value, 532 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86744-3 - Calculus: Third Edition Michael Spivak Index More information Index 663 Function (continued) polynomial, 42 argument, 533 positive part of, 51 discontinuities of, 537 power, 60 complex valued, 532 product of, 43 composition of, 44 quotient of, 43 concave, 217 rational, 42 conjugate, 532 real part, 532 constant, 43 real-valued, 532 continuous, 113 ff. \reasonable", 68, 116, 178 convex, 216 regulated, 515
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