Page 441 - Schaum's Outline of Theory and Problems of Advanced Calculus

432 INDEX

Series (Cont.): Sum, 2 Transforms (see Fourier transforms
geometric, 25, 266 of series, 25, 266 and Laplace transforms)
harmonic, 266 of vectors, 151, 163 Transitivity, law of, 2
Laurent’s, 395, 407, 408, 420 partial, 25, 266 Trigonometric functions, 46, 96
Maclaurin, 274 Summability, 278, 296, 304 derivatives of, 71
of functions of a complex Abel, 305 integrals of, 95, 96
variable, 406–409 Ce ´ saro, 278, 296 inverse, 44
p-, 266 regular, 278, 304 Triple integrals, 210, 219–221
partial sums of, 25, 266 Superior limit (see Limit superior) transformation of, 221–225
power (see Power series) Superposition, principal of, 357 Triple products, scalar, 155
reversion of, 273 Surface, 116 vector, 155
sum of, 25, 266 equipotential, 186
Taylor (see Taylor series) level, 144, 186
telescoping, 278 normal line to (see Normal line) Unbounded interval, 5
terms of, 266 orientable, 248 Uniform continuity, 48, 58, 63, 119
Uniform convergence, 269, 270, 287,
test for integrals, 280 tangent place to (see Tangent
Sets, 1 plane) 288
bounded, 6 Surface integrals, 233–236, 245–249, of integrals, 313, 314
of power series, 272
closed, 6, 12, 13 261 of sequences, 269
connected, 117
countable or denumerable (see of series, 269, 270
Countable set) Tangential component of tests for integrals, 313, 314
elements of, 1 acceleration, 180, 181 tests for series, 270
everywhere dense, 2 Tangent line, to a coordinate curve, theorems for integrals, 314
intersection of, 12 84 theorems for series, 270, 271, 272
orthonormal, 337, 342 to a curve, 65, 184, 202 Weirstrass M test for (see
Weirstrass M test)
point, 117 Tangent plane, 183, 189–191, 200
union of, 12 in curvilinear coordinates, 201, Union of sets, 12
Simple closed curves, 117, 232, 241 202 Unit tangent vector, 157
Unit vectors, 152, 342
Simple poles, 395 Tangent vector, 157, 177
Simply connected region, 117, 232, Taylor polynomials, 273 infinite dimensional, 342
241 Taylor series, in one variable, 274 rectangular, 152
Simpson’s rule, 98, 108, 109 (See also Taylor’s theorem) Upper bound, 6
of functions, 40, 41
Single-valued function, 39, 116, 392 in several variables, 276
Singular points or singularities, of functions of a complex of sequences, 24
395–398, 406–409 variable, 395 Upper limit (see Limit superior)
defined from Laurent series, 395 Taylor’s theorem, 273, 297
essential, 395, 407 (See also Taylor series) Variable, 5, 39
isolated, 395 for functions of one variable, 273 change of, in differentiation, 69, 70
removable, 395, 407 for functions of several variables, change of, in integration, 95,
Sink, 259 276, 277 105–108, 211
Slope, 66 proof of, 297, 407, 408 complex, 392, 393 (See also
Smooth function (see Piecewise remainder in, 274 Functions of a complex
differentiability) Telescoping series, 278 variable)
Solenoidal vector fields, 259 Tensor analysis, 182 dependent and independent, 40,
Source, 259 Term, of a sequence, 23 116
Space curve, 157 of a series, 266 dummy, 94
Specific heat, 356, 357 Terminal point of a vector, 150 limits of integration, 94, 186, 194,
Spherical coordinates, 162, 174, 175 Thermal conductivity, 356, 357 313
arc length element in, 162, 174 Thermodynamics, 148 Vector algebra, 151, 152, 161–165
Laplacian in, 162, 176 Torsion, radius of, 181 Vector analysis (see Vectors)
multiple integrals in, 222 Total differential, 122 (See also Vector:
volume element in, 162, 175 Differentials) bound, 150
Staircase or step function, 51 Trace, on a place, 127 free, 150
Stirling’s asymptotic formula and Transcendental functions, 45, 46 Vector field, 156
series, 378, 384 numbers, 6, 13 solenoidal, 259
Stokes’ theorem, 237, 252–257 Transformations, 124, 139, 140 Vector functions, 156
proof of, 252, 253 and curvilinear coordinates, 139, limits, continuity and derivatives
Stream function, 402 140, 160 of, 156, 171, 172
Subset, 1 conformal, 417 Vector product (see Cross products)
Subtraction, 2 Jacobians of, 125, 160 Vectors, 20, 150–182
of complex numbers, 13, 14 of integrals, 95, 105–108, 211–213, algebra of, 151, 152, 178
of vectors, 151 216–219 axiomatic foundations for, 155