Page 439 - Schaum's Outline of Theory and Problems of Advanced Calculus

430 INDEX

Limits of functions (Cont.): Mean value theorems: Numbers (Cont.):
iterated, 119, 208 for derivatives, 72, 80–82, 87, 125, negative, 1, 2
of a complex variable, 393, 399, 141 operations with, 2–15
400 for integrals, 93, 104, 112 positive, 1, 2
proofs of theorems on, 54–56 Measure zero, 91, 103 rational (see Rational numbers)
right- and left-hand, 45 Mechanics, 158 real (see Real numbers)
special, 46 fluid, 402 roots of, 3
theorems on, 45 Members, of a set, 1 transcendental, 6, 13
Limits of integration, 91 Minimum (see Maxima and Numerator, 1
Limits of sequences, 23, 24, 25, 27 minima) Numerical methods (see
definition of, 23 Moebius strip, 248 Approximations)
of functions, 45, 269 Moment of inertia, 101 for evaluating definite integrals,
theorems of, 23, 24, 28–30 polar, 213, 219 98, 108–110
Limits of vector functions, 156 Monotonic functions, 41
Limit superior, 32, 36 Monotonic sequences, 24, 30–32 Odd functions, 338, 347–351
Linear dependence of vectors, 182 fundamental theorem on, 24
Linear transformations, 148 Multiple integrals, 207–228 Open ball, 117
fractional (see Fractional linear improper, 316 Open interval, 5
region, 117
transformation) in curvilinear coordinates, 211, Operations:
Line integrals, 229–231, 238–240, 212, 221, 222
259 in cylindrical coordinates, 211 with complex numbers, 6, 13, 14
evaluation of, 231 in spherical coordinates, 212 with power series, 372, 373
with real numbers, 2, 8
independence of path of, 232, 238, Jacobians and, 211
243–245 transformations of, 211–213 Ordered pairs of real numbers, 7
properties of, 231 Multiple-valued functions, 39, 117, triplets of real numbers, 155
relation of, to functions of a 392 Order, exponential, 334
of derivatives, 71
complex variable, 394 logarithm as a, 392
vector notation for, 230 Multiplication, 2 of poles, 395, 396
Line, normal (see Normal line) associative law of, 2 Orientable surface, 248
Origin, of a coordinate system, 116
tangent (see Tangent line) involving vectors, 153–155
Logarithms, 4, 10, 11, 351 of complex numbers, 6, 7 Orthogonal curvilinear coordinates
as multiple-valued functions, 392 Multiply-connected regions, 117 (see Curvilinear coordinates)
base of, 4 Orthogonal families, 402, 403
Lower bound, 6, 12, 13 functions, 153, 342, 357, 358
of functions, 40 Natural base of logarithms, 3 Orthonormal functions, 357
of sequences, 24 Natural numbers, 4
Lower limit (see Limit inferior) Negative integers, 1 Pappus’ theorem, 228
L.u.b. (see Least upper bound) numbers, 1, 2 Parabola, 50
Neighborhoods, 6, 117 Parabolic cylindrical coordinates,
Nested intervals, 25, 32 180
Maclaurin series, 274 Newton, Isaac, 65, 90, 265 Parallelepiped, volume of, 155, 169
Magnetic field vector, 181 first and second laws, 68 Parallelogram, area of, 155, 168
Magnitude, of a vector, 150 Newton’s methods, 74 law, 151, 163
Many-valued function (see Normal component of acceleration, Parametric equations, of line, 189
Multiple-valued function) 177 of normal line, 184
Mappings, 124 (See also Normalized vectors and functions, of space curve, 157
Transformations) 342 Parseval’s identity:
conformal, 417 Normal line: for Fourier integrals, 366, 368
Mathematical induction, 8, 15 parametric equations for, 184, 201 for Fourier series, 338, 351, 362,
Maxima and minima, 42, 73, 174, principal, 177, 180 373
185, 187, 196–198 to a surface, 184, 189–191 Partial derivatives, 116–149
absolute, 42 Normal plane, 184, 185, 191, 192 applications of, 183–206
Lagrange’s multiplier method for, nth root test, 268 definition of, 120
188, 198, 199, 204 Null set, 1 evaluation of, 120, 128–130
of functions of several variables, vector, 151 higher order, 120
187, 188, 196–198 Number, cardinal, 5 notations for, 120
relative, 42 Numbers, 1–22 order of differentiation of, 120
Taylor’s theorem and, 276, 277, algebraic (see Algebraic number) using Jacobians, 123
297, 298 Bernoulli, 304 Partial sums of infinite series, 25,
Maximum (see Maxima and complex (see Complex numbers) 265, 266
minima) history, 2, 5 Period, of a function, 336
Maxwell’s equations, 181 irrational (see Irrational numbers) Piecewise continous, 48
Mean square error, 353 natural, 1 differentiable, 66