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INDEX
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Numbers (Cont.): negative, 1, 2 operations with, 2 15 positive, 1, 2 rational (see Rational numbers) real (see Real numbers) roots of, 3 transcendental, 6, 13 Numerator, 1 Numerical methods (see Approximations) for evaluating de nite integrals, 98, 108 110 Odd functions, 338, 347 351 Open ball, 117 Open interval, 5 region, 117 Operations: with complex numbers, 6, 13, 14 with power series, 372, 373 with real numbers, 2, 8 Ordered pairs of real numbers, 7 triplets of real numbers, 155 Order, exponential, 334 of derivatives, 71 of poles, 395, 396 Orientable surface, 248 Origin, of a coordinate system, 116 Orthogonal curvilinear coordinates (see Curvilinear coordinates) Orthogonal families, 402, 403 functions, 153, 342, 357, 358 Orthonormal functions, 357 Pappus theorem, 228 Parabola, 50 Parabolic cylindrical coordinates, 180 Parallelepiped, volume of, 155, 169 Parallelogram, area of, 155, 168 law, 151, 163 Parametric equations, of line, 189 of normal line, 184 of space curve, 157 Parseval s identity: for Fourier integrals, 366, 368 for Fourier series, 338, 351, 362, 373 Partial derivatives, 116 149 applications of, 183 206 de nition of, 120 evaluation of, 120, 128 130 higher order, 120 notations for, 120 order of di erentiation of, 120 using Jacobians, 123 Partial sums of in nite series, 25, 265, 266 Period, of a function, 336 Piecewise continous, 48 di erentiable, 66
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INDEX
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Product, 1 box, 155 cross or vector (see Cross products) dot or scalar (see Dot products) in nite (see In nite product) nth derivative of, 89 triple (see Triple products) Wallis , 359 p series, 266 Quadratic equation, solutions of, 14 Quadratic form, 206 Quotient, 1 Quotient test: for integrals, 309, 311, 317 for series, 267, 279, 280 Raabe s test, 268, 285 Radius of convergence, 272, 276 of curvature, 177, 181 of torsion, 181 Radius vector, 153 Range, of integration, 91 Rates of change, 74 Rational algebraic functions, 43 Rational numbers, 1, 9, 10 countability of, 11, 12 Ratio test, 268, 284, 285 proof of, 284 Real axis, 2 Real numbers, 1 (See also Numbers) absolute value of, 3 axiomatic foundations of, 3 decimal representation of, 2 geometric representation of, 2 inequalities for (see Inequality) non-countability of, 12 operations with, 2, 8, 9 ordered pairs and triplets of, 7, 155 roots of, 3, 11 Real part: of a complex number, 6 of functions of a complex variable, 392, 399, 400 Rectangular component vectors, 152 Rectangular coordinates, 7, 116, 160 Rectangular neighborhood, 117 rule for integration, 98 Recurring: decimal, 2 Region, 117 closed, 117 connected, 232 multiply-connected, 117 of convergence, 117 open, 117 simply-connected, 117, 232, 241 Regular summability, 278, 304 Relative extrema, 73 Relativity, theory of, 182 Removable discontinuity, 56, 119 singularity, 393, 407 Residues, 397, 409 412 Residue theorem, 397, 398, 409 412 evaluation of integrals by, 398, 403 406 proof of, 409, 410 Resultant of vectors, 151, 163 Reversion of series, 273 Riemann: axis, 396 surface, 397 Riemann integrable, 91 Riemann s theorem, 354, 370 Right-hand continuity, 47 derivatives, 67, 77, 78 limits, 45 Right-handed rectangular coordinate system, 152, 153 Rolle s theorem, 72 proof of, 80 Roots: of complex numbers, 7, 14 of real numbers, 3, 11 Roots of equations, 43 computations, 59 Newton s method for nding, 89 Saddle points, 188 Scalar, 153 eld, 153 invariant, 182 product (see Dot products) triple product, 155 Scale factors, 160 Scale of two (see Binary scale) Schwarz s inequality: for integrals, 110 for real numbers, 10, 18 Section (see Dedekind cut) Separation of variables in boundaryvalue problems, 356 Sequence, Fibonacci, 35 Sequences, 23 38, 269 bounded, monotonic, 24, 30 32 convergent and divergent, 23, 269 decreasing, 25 de nition of, 23 nite and in nite, 269 increasing, 25 limits of, 23, 27, 269 (See also Limits of sequences) of functions, 269 terms of, 26 uniform covergence of, 269 Series (see In nite series) alternating (see Alternating series) asymptotic (see Asymptotic series) binomial, 275 double, 277
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p integrals, 308 Plane, complex, 7 Plane, equation of, 170 normal to a curve (see Normal plane) tangent to a surface (see Tangent place) Point: boundary, 117 branch, 396, 397 cluster, 5, 117 (See also Limit points) critical, 73 interior, 117 limit (see Limit points) neighborhood of, 5, 117 of accumulation, 5 (See also Limit points) singular (see Singular points) Point set: one-dimensional, 5 two-dimensional, 117 Polar coordinates, 7 Polar form, of complex numbers, 7, 14 Poles, 395 de ned from a Laurent series, 395 of in nite order, 395 residues at, 395 Polynomial functions, 43 degree of, 43 Position vector, 157 Positive de nite quadratic form, 206 Positive direction, 232 normal, 236 Positive integers, 1 numbers, 1, 2 Potential, velocity, 402 Power series, 272, 275, 276, 291 294 Abel s theorem on, 272 expansion of functions in, 273 operations with, 273, 274 radius of convergence of, 272 special, 276, 277 theorems on, 272 uniform covergence of, 272 Prime, relatively, 9 Principal branch: of a function, 41 of a logarithm, 397 Principal normal, to a space curve, 177, 180 Principal part, 67, 120 of a Laurent series, 395 Principal value: of functions, 41, 44, 45 of integrals (see Cauchy principal value) of inverse hyperbolic functions, 44 of inverse trigonometric functions, 44 of logarithms, 392
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Series (Cont.): geometric, 25, 266 harmonic, 266 Laurent s, 395, 407, 408, 420 Maclaurin, 274 of functions of a complex variable, 406 409 p-, 266 partial sums of, 25, 266 power (see Power series) reversion of, 273 sum of, 25, 266 Taylor (see Taylor series) telescoping, 278 terms of, 266 test for integrals, 280 Sets, 1 bounded, 6 closed, 6, 12, 13 connected, 117 countable or denumerable (see Countable set) elements of, 1 everywhere dense, 2 intersection of, 12 orthonormal, 337, 342 point, 117 union of, 12 Simple closed curves, 117, 232, 241 Simple poles, 395 Simply connected region, 117, 232, 241 Simpson s rule, 98, 108, 109 Single-valued function, 39, 116, 392 Singular points or singularities, 395 398, 406 409 de ned from Laurent series, 395 essential, 395, 407 isolated, 395 removable, 395, 407 Sink, 259 Slope, 66 Smooth function (see Piecewise di erentiability) Solenoidal vector elds, 259 Source, 259 Space curve, 157 Speci c heat, 356, 357 Spherical coordinates, 162, 174, 175 arc length element in, 162, 174 Laplacian in, 162, 176 multiple integrals in, 222 volume element in, 162, 175 Staircase or step function, 51 Stirling s asymptotic formula and series, 378, 384 Stokes theorem, 237, 252 257 proof of, 252, 253 Stream function, 402 Subset, 1 Subtraction, 2 of complex numbers, 13, 14 of vectors, 151 Sum, 2 of series, 25, 266 of vectors, 151, 163 partial, 25, 266 Summability, 278, 296, 304 Abel, 305 Cesaro, 278, 296 regular, 278, 304 Superior limit (see Limit superior) Superposition, principal of, 357 Surface, 116 equipotential, 186 level, 144, 186 normal line to (see Normal line) orientable, 248 tangent place to (see Tangent plane) Surface integrals, 233 236, 245 249, 261 Tangential component of acceleration, 180, 181 Tangent line, to a coordinate curve, 84 to a curve, 65, 184, 202 Tangent plane, 183, 189 191, 200 in curvilinear coordinates, 201, 202 Tangent vector, 157, 177 Taylor polynomials, 273 Taylor series, in one variable, 274 (See also Taylor s theorem) in several variables, 276 of functions of a complex variable, 395 Taylor s theorem, 273, 297 (See also Taylor series) for functions of one variable, 273 for functions of several variables, 276, 277 proof of, 297, 407, 408 remainder in, 274 Telescoping series, 278 Tensor analysis, 182 Term, of a sequence, 23 of a series, 266 Terminal point of a vector, 150 Thermal conductivity, 356, 357 Thermodynamics, 148 Torsion, radius of, 181 Total di erential, 122 (See also Di erentials) Trace, on a place, 127 Transcendental functions, 45, 46 numbers, 6, 13 Transformations, 124, 139, 140 and curvilinear coordinates, 139, 140, 160 conformal, 417 Jacobians of, 125, 160 of integrals, 95, 105 108, 211 213, 216 219
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