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| Wrede R.C., Spiegel M. — Theory and problems of advanced calculus |
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| Предметный указатель |
Intervals, open 5
Intervals, unbounded 5
Invariance relations 181 182
Invariant, Fourier transforms 369 see
Invariant, Laplace transforms 315 423 see
Invariant, scalar 182
Inverse functions 41
Inverse functions, continuity of 47
Inverse functions, hyperbolic 45
Inverse functions, trigonometric 44
Inverse, of addition and multiplication 2 3
Irrantional algebraic functions 43
Irrational numbers 2 9 10
Irrational numbers, approximations to 9
Irrational numbers, definition of 2 see
Irrationality of  , proof of 9
Isolated singularity 395
Iterated integrals 208—210
Iterated limits 119
Jacobian determinant see "Jacobians"
Jacobians 123 135—139 161 162 174 175
Jacobians, chain rules for 124
Jacobians, curvilinear coordinates and 161 162
Jacobians, functions of a complex variable and 422
Jacobians, implicit functions and 135—139
Jacobians, multiple integrals and 211
Jacobians, of transformations 124
Jacobians, partial derivatives using 123
Jacobians, theorems on 124 162
Jacobians, vector interpretation of 160
Kronecker's symbol 342
L'Hospital's rules 72 82—84 88
L'Hospital's rules, proofs of 82 83
l.u.b. see "Least upper bound"
Lagrange multipliers 188 198 199
Lagrange's form of the remainder, in Taylor series 274 297
Laplace transforms 314 315 333
Laplace transforms, convolution theorem for 334
Laplace transforms, inverse 330 423
Laplace transforms, relation of functions of a complex variable to 423
Laplace transforms, table of 315
Laplace transforms, use of, in solving differential equations 315 330
Laplace's equation 129 see
Laplacian operator 161 162 see
Laplacian operator, in curvilinear coordinates 161
Laplacian operator, in cylindrical coordinates 161
Laplacian operator, in spherical coordinates 161
Laurent's series 395 407 408
Laurent's series, theorem 408 409
Least limit see "Limit inferior"
Least square approximations 201
Least upper bound 6 32
Least upper bound of functions 41
Least upper bound of sequences 24 36
Left-hand continuity 47
Left-hand derivatives 67 77 78
Left-hand limits 45
Leibnitz's formula for nth derivative of a product 89
Leibnitz's rule for differentiating under the integral sign 186 194
Leibniz, Gottfried Wilhelm 65 90 265
Lemniscate 114
Length, of a vector 150
Less than 2
Level curves and surfaces 144 186
Limit inferior 32 36
Limit points 5 12 117
Limit points, Bolzano — Weirstrass theorem on see "Bolzano — Weirstrass"
Limit superior 32 36
limits of functions 39—64 117 118 269 393 399 400
Limits of functions, definition of 43 118 119
Limits of functions, iterated 119 208
Limits of functions, of a complex variable 393 399 400
Limits of functions, proofs of theorems on 54—56
Limits of functions, right- and left-hand 45
Limits of functions, special 46
Limits of functions, theorems on 45
Limits of integration 91
Limits of sequences 23 24 25 27
Limits of vector functions 156
Limits, definition of 23
Limits, theorems of 23 24 28—30
Line integrals 229—231 238—240 259
Line integrals, evaluation of 231
Line integrals, independence of path of 232 238 243—245
Line integrals, properties of 231
Line integrals, relation of, to functions of a complex variable 394
Line integrals, vector notation for 230
Linear dependence of vectors 182
Linear transformations 148
Logarithms 4 10 11 351
Logarithms, as multiple-valued functions 392
Logarithms, base of 4
Lower bound 6 12 13
Lower bound, of functions 40
Lower bound, of sequences 24
Lower limit see "Limit inferior"
Maclaurin series 274
Magnetic field vector 181
Magnitude, of a vector 150
Many-valued function see "Multiple-valued function"
Mappings 124 see
Mappings, conformal 417
Mathematical induction 8 15
Maxima and minima 42 73 174 185 187 196—198
Maxima and minima, absolute 42
Maxima and minima, Lagrange's multiplier method for 188 198 199 204
Maxima and minima, of functions of several variables 187 188 196—198
Maxima and minima, relative 42
Maxima and minima, Taylor's theorem and 276 277 297 298
Maximum see "Maxima and minima"
Maxwell's equations 181
Mean square error 353
Mean value theorems, for derivatives 72 80—82 87 125 141
Mean value theorems, for integrals 93 104 112
Measure zero 91 103
Mechanics 158
Mechanics, fluid 402
Members, of a set 1
Minimum see "Maxima and minima"
Moebius strip 248
Moment of inertia 101
Moment of Inertia, Polar 213 219
Monotonic functions 41
Monotonic sequences 24 30—32
Monotonic sequences, fundamental theorem on 24
Multiple integrals 207—228
Multiple integrals, improper 316
Multiple integrals, in curvilinear coordinates 211 212 221 222
Multiple integrals, in cylindrical coordinates 211
Multiple integrals, in spherical coordinates 212
Multiple integrals, Jacobians and 211
Multiple integrals, transformations of 211—213
Multiple-valued functions 39 117 392
Multiple-valued functions, logarithm as a 392
Multiplication 2
Multiplication, associative law of 2
Multiplication, involving vectors 153—155
Multiplication, of complex numbers 6 7
Multiply-connected regions 117
Natural base of logarithms 3
Natural numbers 4
Negative integers 1
negative numbers 1 2
Neighborhoods 6 117
Nested intervals 25 32
Newton's methods 74
Newton, first and second laws 68
Newton, Isaac 65 90 265
Normal component of acceleration 177
Normal line, parametric equations for 184 201
| Normal line, principal 177 180
Normal line, to a surface 184 189—191
Normal plane 184 185 191 192
Normalized vectors and functions 342
nth root test 268
Null set 1
Null vector 151
Number, cardinal 5
Numbers 1—22
Numbers, Bernoulli 304
Numbers, history 2 5
Numbers, natural 1
numbers, negative 1 2
Numbers, operations with 2—15
Numbers, positive 1 2
Numbers, roots of 3
Numbers, transcendental 6 13
Numerator 1
Numerical methods see "Approximations"
Numerical methods, for evaluating definite integrals 98 108—110
Odd functions 338 347—351
Open ball 117
Open interval 5
Open region 117
Operations, with complex numbers 6 13 14
Operations, with power series 372 373
Operations, with real numbers 2 8
Order, exponential 334
Order, of derivatives 71
Order, of poles 395 396
Ordered pairs of real numbers 7
Ordered triplets of real numbers 155
Orientable surface 248
Origin, of a coordinate system 116
Orthogonal curvilinear coordinates see "Curvilinear coordinates"
Orthogonal families 402 403
Orthogonal functions 153 342 357 358
Orthonormal functions 357
p integrals 308
p series 266
P-series 266
Pappus' theorem 228
parabola 50
Parabolic cylindrical coordinates 180
Parallelepiped, volume of 155 169
Parallelogram, area of 155 168
Parallelogram, law 151 163
Parametric equations, of line 189
Parametric equations, of normal line 184
Parametric equations, of space curve 157
Parseval's identity, for Fourier integrals 366 368
Parseval's identity, for Fourier series 338 351 362 373
Partial derivatives 116—149
Partial derivatives, applications of 183—206
Partial derivatives, definition of 120
Partial derivatives, evaluation of 120 128—130
Partial derivatives, higher order 120
Partial derivatives, notations for 120
Partial derivatives, order of differentiation of 120
Partial derivatives, using Jacobians 123
Partial sums of infinite series 25 265 266
Period, of a function 336
Piecewise continous 48
Piecewise differentiable 66
Plane, complex 7
Plane, equation of 170
Plane, normal to a curve see "Normal plane"
Plane, tangent to a surface see "Tangent place"
Point set, one-dimensional 5
Point set, two-dimensional 117
Point, boundary 117
Point, branch 396 397
Point, cluster 5 117 see
Point, critical 73
Point, interior 117
Point, neighborhood of 5 117
Point, of accumulation 5 see
Polar coordinates 7
Polar form, of complex numbers 7 14
Poles 395
Poles, defined from a Laurent series 395
Poles, of infinite order 395
Poles, residues at 395
Polynomial functions 43
Polynomial functions, degree of 43
Position vector 157
Positive definite quadratic form 206
Positive direction 232
Positive direction, normal 236
Positive integers 1
Positive numbers 1 2
Potential, velocity 402
Power series 272 275 276 291—294
Power series, Abel's theorem on 272
Power series, expansion of functions in 273
Power series, operations with 273 274
Power series, radius of convergence of 272
Power series, special 276 277
Power series, theorems on 272
Power series, uniform convergence of 272
Prime, relatively 9
Principal branch, of a function 41
Principal branch, of a logarithm 397
Principal normal, to a space curve 177 180
Principal part 67 120
Principal part of a Laurent series 395
Principal value, of functions 41 44 45
Principal value, of integrals see "Cauchy principal value"
Principal value, of inverse hyperbolic functions 44
Principal value, of inverse trigonometric functions 44
Principal value, of logarithms 392
PRODUCT 1
Product, box 155
Product, cross or vector see "Cross products"
Product, dot or scalar see "Dot products"
Product, infinite see "Infinite product"
Product, nth derivative of 89
Product, Wallis' 359
Quadratic equation, solutions of 14
Quadratic form 206
Quotient 1
Quotient test, for integrals 309 311 317
Quotient test, for series 267 279 280
Raabe's test 268 285
Radius of convergence 272 276
Radius of curvature 177 181
Radius of torsion 181
Radius vector 153
Range, of integration 91
Rates of change 74
Ratio Test 268 284 285
Ratio test, proof of 284
Rational algebraic functions 43
Rational numbers 1 10
Rational numbers, countability of 11 12
Real axis 2
Real numbers 1 see
Real numbers, absolute value of 3
Real numbers, axiomatic foundations of 3
Real numbers, decimal representation of 2
Real numbers, geometric representation of 2
Real numbers, inequalities for see "Inequality"
Real numbers, non-countability of 12
Real numbers, operations with 2 8 9
Real numbers, ordered pairs and triplets of 7 155
Real numbers, roots of 3 11
Real part, of a complex number 6
Real part, of functions of a complex variable 392 399 400
Rectangular component vectors 152
Rectangular coordinates 7 116 160
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