Page 435 - Schaum's Outline of Theory and Problems of Advanced Calculus

426 INDEX

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