Page 440 - Schaum's Outline of Theory and Problems of Advanced Calculus

INDEX 431

p integrals, 308 Product, 1 Relativity, theory of, 182
Plane, complex, 7 box, 155 Removable discontinuity, 56, 119
Plane, equation of, 170 cross or vector (see Cross singularity, 393, 407
normal to a curve (see Normal products) Residues, 397, 409–412
plane) dot or scalar (see Dot products) Residue theorem, 397, 398, 409–412
tangent to a surface (see Tangent infinite (see Infinite product) evaluation of integrals by, 398,
place) nth derivative of, 89 403–406
Point: triple (see Triple products) proof of, 409, 410
boundary, 117 Wallis’, 359 Resultant of vectors, 151, 163
branch, 396, 397 p series, 266 Reversion of series, 273
cluster, 5, 117 (See also Limit Riemann:
points) axis, 396
critical, 73 Quadratic equation, solutions of, 14 surface, 397
interior, 117 Quadratic form, 206 Riemann integrable, 91
limit (see Limit points) Quotient, 1 Riemann’s theorem, 354, 370
Quotient test:
neighborhood of, 5, 117 Right-hand continuity, 47
of accumulation, 5 (See also Limit for integrals, 309, 311, 317 derivatives, 67, 77, 78
points) for series, 267, 279, 280 limits, 45
singular (see Singular points) Right-handed rectangular
Point set: Raabe’s test, 268, 285 coordinate system, 152, 153
one-dimensional, 5 Radius of convergence, 272, 276 Rolle’s theorem, 72
two-dimensional, 117 of curvature, 177, 181 proof of, 80
Polar coordinates, 7 of torsion, 181 Roots:
Polar form, of complex numbers, 7, Radius vector, 153 of complex numbers, 7, 14
14 Range, of integration, 91 of real numbers, 3, 11
Poles, 395 Rates of change, 74 Roots of equations, 43
defined from a Laurent series, 395 Rational algebraic functions, 43 computations, 59
of infinite order, 395 Rational numbers, 1, 9, 10 Newton’s method for finding, 89
residues at, 395 countability of, 11, 12
Polynomial functions, 43 Ratio test, 268, 284, 285
degree of, 43 proof of, 284 Saddle points, 188
Position vector, 157 Real axis, 2 Scalar, 153
Positive definite quadratic form, Real numbers, 1 (See also Numbers) field, 153
206 absolute value of, 3 invariant, 182
Positive direction, 232 axiomatic foundations of, 3 product (see Dot products)
normal, 236 decimal representation of, 2 triple product, 155
Positive integers, 1 geometric representation of, 2 Scale factors, 160
numbers, 1, 2 inequalities for (see Inequality) Scale of two (see Binary scale)
Potential, velocity, 402 non-countability of, 12 Schwarz’s inequality:
Power series, 272, 275, 276, 291–294 operations with, 2, 8, 9 for integrals, 110
Abel’s theorem on, 272 ordered pairs and triplets of, 7, for real numbers, 10, 18
expansion of functions in, 273 155 Section (see Dedekind cut)
operations with, 273, 274 roots of, 3, 11 Separation of variables in boundary-
radius of convergence of, 272 Real part: value problems, 356
special, 276, 277 of a complex number, 6 Sequence, Fibonacci, 35
theorems on, 272 of functions of a complex Sequences, 23–38, 269
uniform covergence of, 272 variable, 392, 399, 400 bounded, monotonic, 24, 30–32
Prime, relatively, 9 Rectangular component vectors, convergent and divergent, 23, 269
Principal branch: 152 decreasing, 25
of a function, 41 Rectangular coordinates, 7, 116, 160 definition of, 23
of a logarithm, 397 Rectangular neighborhood, 117 finite and infinite, 269
Principal normal, to a space curve, rule for integration, 98 increasing, 25
177, 180 Recurring: limits of, 23, 27, 269 (See also
Principal part, 67, 120 decimal, 2 Limits of sequences)
of a Laurent series, 395 Region, 117 of functions, 269
Principal value: closed, 117 terms of, 26
of functions, 41, 44, 45 connected, 232 uniform covergence of, 269
of integrals (see Cauchy principal multiply-connected, 117 Series (see Infinite series)
value) of convergence, 117 alternating (see Alternating series)
of inverse hyperbolic functions, 44 open, 117 asymptotic (see Asymptotic
of inverse trigonometric simply-connected, 117, 232, 241 series)
functions, 44 Regular summability, 278, 304 binomial, 275
of logarithms, 392 Relative extrema, 73 double, 277