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Creation QR Code 2d barcode in .NET Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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Cauchy-Riemann equations (Cont.): in polar form, 422 Cauchy s: convergence criterion, 25, 33 form of remainder in Taylor s theorem, 274, 296 generalized theorem of the mean, 72, 82 inequality, 422 integral formulas, 394, 403 406 theorem, 394, 403 406 Centripetal acceleration, 175 Chain rules, 69, 122, 133 for Jacobians, 124 Circle of convergence, 276 Class, 1 (See also Sets) Closed interval, 5 region, 117 set, 6, 12, 13, 117 Closure law or property, 2 Cluster point, 5, 117 (See also Limit points) Collection, 1 (See also Sets) Commutative law, 2 for dot products, 153 for vectors, 154, 166, 167 Comparison test: for integrals, 308, 311, 319 for series, 267, 279, 280 Completeness, of an orthonormal set, 310 Complex numbers, 6, 13, 14 absolute value of, 6 amplitude of, 7 argument of, 7 as ordered pairs of real numbers, 7 as vectors, 20 axiomatic foundations of, 7 conjugate of, 6 equality of, 6 modulus of, 6 operations with, 6, 13, 14 polar form of, 7, 14 real and imaginary parts of, 6 roots of, 7, 13 Complex plane, 7 Complex variable, 392, 393 (See also Functions of a complex variable) Components, of a vector, 153 Composite fuctions, 47 continuity of, 47 di erentiation of, 69, 132 135 Conditional convergence: of integrals, 309, 312, 319, 320 of series, 268, 300 Conductivity, thermal, 357 Conformal mapping or transformation, 417 (See also Transformations) Conjugate, complex 6 Connected region, 237 set, 117 Connected region (Cont.): simply-, 117, 232 Conservative eld, 233 Constants, 5 Constraints, 188 Continuity, 46 64, 119, 127, 128, 399 and di erentiability, 66, 72, 73, 120, 121 de nition of, 46, 47 in an interval, 47 in a region, 119 of an in nite series of functions, 271, 272, 288 of functions of a complex variable, 393, 399, 400 of integrals, 99, 314 of vector functions, 156 right- and left-hand, 47 piecewise, 48 theorems on, 47, 48 uniform, 48, 119 Continuous (see Continuity) di erentiability, 67, 121 Continuously di erentiable functions, 66, 67, 120 Continuum, cardinality of, 5 Contour integration, 398 Convergence: absolute (see Absolute convergence) circle of, 276 conditional (see Conditional convergence) criterion of Cauchy, 33, 37 domain of, 272 interval of, 25, 272 of Fourier integrals (see Fourier s integral theorem) of Fourier series, 338, 354 356 of improper integrals (see Improper integrals) of in nite series (see In nite series) of series of constants, 278 285 radius of, 272, 276 region of, 117 uniform (see Uniform convergence) Convergent (see Convergence) integrals, 306 309 (See also Improper integrals) sequences, 23, 269 (See also Sequences) series, 25 (See also In nite series) Convolution theorem: for Fourier transforms, 365 for Laplace transforms, 334 Coordinate curve, 160 Coordinates: curvilinear, 139, 160 (See also Curvilinear coordinates) cylindrical, 161, 174 hyperbolic, 218
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Coordinates (Cont.): polar, 7 rectangular, 152 spherical, 162, 190 Correspondence, 2, 11, 23, 39, 160 one to one, 2, 11 Countability, 5, 11, 12 of algebraic numbers, 13 of rational numbers, 11, 12 Countable set, 5, 11, 12 measure of a, 91, 103 Critical points, 73 Cross products, 154, 166 169 proof of distributive law for, 166 Curl, 158, 159, 172 174 in curvilinear coordinates, 161 Curvature, radius of, 177, 181 Curve, coordinate, 150 simple closed, 117, 232, 242 space, 157 Curvilinear coordinates, 125, 139 curl, divergence, gradient, and Laplacian in, 161, 162 Jacobians and, 161, 162 multiple integrals in, 207 228 orthogonal, 207 228 special, 161, 162 transformations and, 139, 140, 160 vectors and, 161, 162 Cut (see Dedekind cut) Cycloid, 99 Cylindrical coordinates, 161, 174, 175 arc length element in, 161 divergence in, 175 gradient in, 175 Laplacian in, 161, 173 multiple integrals in, 222 parabolic, 180 volume element in, 161, 175 Decimal representation of real numbers, 2 Decimals, recurring, 2 Decreasing functions, 41, 47 monotonic, 41 strictly, 41, 47 Decreasing sequences, monotonic and strictly, 24 Dedekind cuts, 4, 16 De nite integrals, 90 95, 103 (See also Integrals) change of variable in, 95, 105 108 de nition of, 90, 91 mean value theorems for, 92, 93, 104 numerical methods for evaluating, 98, 108, 109 properties of, 91, 92 theorem for existence of, 91 with variable limits, 95, 186, 313, 314
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