Widder D.V. — Advanced calculus :: Электронная библиотека попечительского совета мехмата МГУ

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Widder D.V. — Advanced calculus

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Название: Advanced calculus

Автор: Widder D.V.

Аннотация:

Classic text leads from elementary calculus into more theoretic problems. Precise approach with definitions, theorems, proofs, examples and exercises. Topics include partial differentiation, vectors, differential geometry, Stieltjes integral, infinite series, gamma function, Fourier series, Laplace transform, much more. Numerous graded exercises with selected answers.

Язык:

Рубрика: Математика/Анализ/Учебники по элементарному анализу/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1947

Количество страниц: 432

Добавлена в каталог: 18.04.2005

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Предметный указатель

, definition of class       7 11
, definition of class       330
(C,1)      see Cesaro summability
Abel summability      205 301
Abscissa: of absolute convergence for a Laplace transform      374
Abscissa: of convergence for a Laplace transform      371
Adams, J.C.      307
Angle between surface curves      91
Approximation: least square      354 350
Approximation: polynomial      355
Approximation: trigonometric      354
Arc length      90
Area: of a region      184 193
Area: of a surface      172
Arithmetic means      see Cesaro summability
Attraction for a continuous distribution of mass      174 178
Bessel's function of order n      391
Bessel's inequality      333 334 337 357
Beta function, B(x, y)      308 ff 311
Bilateral convolution      300
Bilateral Laplace transform      395 398
Bilateral resultant      300
Bonnet's theorem, Weierstrass form of      138
Bound: greatest lower      149
Bound: least upper      147 149
Bound: of continuous function      146
Boundary-value problem      346
Bounded variation of a function      132
C, definition of class C      6 10
Cauchy remainder in Taylor's series      35
Cauchy's convergence test      242
Cauchy's criterion      233 235
Center of gravity      168 178
Cesaro summability      202—205 300 351
Class C, D, , E, P, etc.      see C D E P etc.
Comparison tests for convergence and divergence: of improper integrals      209 280 281
Comparison tests for convergence and divergence: of infinite series      239 240
Components of a vector      50
Contact of order n      76
Continuity, uniform: for functions of a single variable      147
Continuity, uniform: for functions of two variables      183
Continuous distribution of mass      142 174
Continuous function      6 10 146—149 291 296
Convergence theorem for Fourier series      330
Convolution      380 see
Cosine integral, CI(s)      383 385 400 404
Cramer's rule      21
Cross derivatives, equality of      41—43
Curl      65
Curvature      80
Curvature, mean      93
Curvature, normal      92
Curvature, total, or Gaussian      03
Curve in space: arc length of      72
Curve in space: normal plane to      74
Curve in space: tangent line to      73
Curves: closed      180
Curves: family of plane      118
Curves: Jordan      184 180 187
Curves: regular      180
Curves: sectionally smooth      180
Curvilinear integral      see Integral
Cylindrical coordinates for triple integrals      179
D'Alembert's ratio tent      242
D, definition of class D      330
Definite integrals, evaluation of      313—315
Del ( $\nabla$ )      65
Delta-neighborhood ( $\delta$ -neighborhood)      10
Density of a continuous distribution      142
Derivative: directional      30 66
Derivative: of higher order      3 16 20
Derivative: on the left      6
Derivative: on the right      6
Derivatives: cross, equality of      41—43
Derivatives: partial, definitions of      1 11
Determining function      402
Determining function, definition of      367
Determining sequence      417
Developable surface      123
Developable surface, polar      124
Developable surface, rectifying      124
Diameter of a region      154
Dienes, P.      180
Difference equations      410
Differential equation: exact      196
Differential equation: homogeneous linear      408
Differential equation: linear      404
Differential equation: nonhomogeneous linear      412
Differential equation: of hyperbolic typo      346
Differential equation: of vibrating string      344
Differential equation: partial      420
Differential: definition of      20
Differential: exact      213
Differential: existence of exact      195
Differentiation: of a function defined by a proper integral      291—293
Differentiation: of a function defined by an improper integral      298 313
Differentiation: of implicit functions      10
Differentiation: of in finite series      259
Differentiation: partial      1—19
Directional derivative      30 66
Dirichlet formula      166 100
Dirichlet series      300 360 397
Divergence, Div      65
Divergent integrals      300—302
Divergent series      201—265
Domain      10
Domain, bounded      153
Double integral: applications of      108
Double integral: definition of      154
Double integral: existence of      182
Double integral: iteration of; in polar coordinates      102—104
Double integral: iteration of; in rectangular coordinates      159—161
Double integral: properties of      156
Duhamel's theorem      72 132 144 147 168 181 183 185 188 190 200
E, definition of class E      409
Eider's constant      307 321
Eider's theorem      15
Envelope      118
Envelope, of families of surfaces      122
Envelope, of normals      120
Envelope, of tangents      120
Equations: difference      416
Equations: simultaneous      21 47
Error function (err)      383 385 399
Exponential integral, EI(x)      383 384 400 401 403
extrema      105. See also Maximum and minimum
Faltung      380. See also Resultant
Fejer's theorem      352 354 355
Fejer, L.      351
Field of force, conservative      190
First fundamental form of a surface      90
First mean-value theorem for Riemann integrals      34
First mean-value theorem for Stieltjes integrals      137
Fourier coefficients      324 327
Fourier cosine transform      302
Fourier integral      324 359
Fourier integral, convergence theorem for      302
Fourier series: convergence of      333
Fourier series: convergence theorem for      336
Fourier series: definition of      324
Fourier series: for an arbitrary interval      343
Fourier series: summability of      351
Fourier sine transform      362
Fourier transform      302
Frenet — Serret formulas      83 92 124
Function: Bessel's, of order n      394
Function: beta      308 341
Function: composite      12
Function: continuous      6 10 145—149
Function: defined by a proper integral: continuity of      291 292

Function: defined by a proper integral: differentiation of      291—293
Function: defined by a proper integral: integration of      159 162 165 166
Function: defined by an improper integral: continuity of      296
Function: defined by an improper integral: differentiation of      208 313
Function: defined by an improper integral: integration of      297
Function: determining: differentiation of      374
Function: determining: product of      380
Function: error, (erf)      383 385 399
Function: even      327
Function: gamma      303 341 308 398 300
Function: generating      see Generating function
Function: homogeneous      14—10
Function: implicit      2 18 38 44
Function: integrator      120
Function: limit of a      4 9
Function: multiple-valued, single-valued      1
Function: normed      325
Function: odd      327
Function: of bounded variation      132
Function: of class       7 11
Function: of class C      6 10
Function: of one variable      4—8
Function: of several variables      6—13
Function: uniformly continuous: in a region      183
Function: uniformly continuous: on an interval      147
Functional dependence      45
Functions, orthogonal      325
Fundamental form of a surface: first      90
Fundamental form of a surface: second      91
Gamma function, $\Gamma(x)$       303 ff 341 308 398 399
Gauss's theorem      101 197 307
Generating function: definition of      307
Generating function: differentiation of      374
Generating function: of a sequence      417
Generating functions, product of      380
Gradient, Grad      32 05 00
Gravitational attraction      see Attraction
Greatest lower bound      149
Green's theorem: in three dimensions      200—202 206 213—215
Green's theorem: in two dimensions      191—199 210
Gyration, radius of      171
Harmonic analysis      343 350
Harmonic series      240
Heine — Borel theorem      115 140 185
Helix      58
Hoelder summability      205
Homogeneous function      14
Homogeneous linear differential equation      408—412
Homogeneous polynomial      3
Hyperbolic type, differential equation of      340
Implicit function theorem      44
Improper integrals: absolute convergence of      270 281 285
Improper integrals: classification of      267 285 280
Improper integrals: comparison teats for      209 280
Improper integrals: conditional convergence of      270 276—278
Improper integrals: convergence of Type I      208
Improper integrals: convergence of Type III      280
Improper integrals: convergence of Types II and IV      284
Improper integrals: divergence of      208 285 300—302
Improper integrals: functions defined by      see Function
Improper integrals: limit tests for      273 281 285
Improper integrals: summability of divergent      300—302
Improper integrals: uniform convergence of      288—290 296—299
Improper integrals: Weierstrass M-test for uniform convergence of      280 302 375
Indeterminant form: evaluation by orders of infinity      230
Indeterminant form: evaluation by series      228
Indeterminant form: L'Hospital's rule      218 221 225 227
Indeterminant form: type       210
Indeterminant form: type $\infty/\infty$       220
Indeterminant form: types , $\infty^0$ , $1^\infty$       220
Indeterminant form: types $\infty-\infty$ , $0\cdot\infty$       225
Inertia, moment of      120 142 170
Inferior limit      233
Infinite series: absolute convergence of      245—240 255
Infinite series: alternating      245—247
Infinite series: comparison tests for      230—240
Infinite series: conditional convergence of      240
Infinite series: continuity of the sum of an      257
Infinite series: convergence tests for      239 240 242—244 240—251 254
Infinite series: definition of convergence, and divergence of      298
Infinite series: differentiation of      250—201
Infinite series: integration of      257—250
Infinite series: method of, for evaluation of definite integrals      315
Infinite series: summation of divergent      201—205
Infinite series: uniform convergence of      252—255
Infinitesimal      231
Infinities, table of      230
Infinity, definition of orders of      230
Integral: change of variable in      204
Integral: curvilinear      185
Integral: divergent      300—302
Integral: double      see Double integral
Integral: Fourier      324 359 362
Integral: improper      see improper integral
Integral: iterated      157 165 176
Integral: Lebesgue      185
Integral: line: in a plane      180—198
Integral: line: in space      210—214
Integral: Maclaurin's integral test for infinite series      243
Integral: multiple      see Double integral Triple
Integral: proper: function defined by      see Function
Integral: proper: properties of      291—294
Integral: proper: Riemann      see Riemann integral
Integral: proper: Stieltjes      see Stieltjes integral
Integral: proper: surface      186 199—204
Integral: proper: triple      see Triple integral
Integrals, definite; evaluation of      313—315
Integration: by parts      134
Integration: change in order of      105—107 297
Integration: of a function defined by a proper integral      159 102 165 166
Integration: of a function defined by an improper integral      297
Integration: of infinite series      257
Integrator function      120
Invariants      68—70
Jacobian      22 38 200 208 209
Jordan curves      184 180 187
Just scale      349
l'Hospital's rule      218 221 225 227
Lagrange identity      51 108
Lagrange remainder      35
Lagrange's multipliers      113
Laguerre polynomial,       384 385 390
Lamina: attraction of, on a unit particle      174
Lamina: center of gravity of      168
Lamina: moment of inertia of      144
Lapiace — Stieltjes transform      307
Laplace transform      365—423
Laplace transform table of      399
Laplace transform, bilateral      395 398
Laplace transform, definition of      314 307
Laplace transform, inversion formula for      390
Laplace transform, operational properties of      370
Laplace transform, unilateral      305
Laplace's method      21
Laplacian, ( $\nabla\big|\nabla$ )      65
Law of the mean      8 72 2J0
Law of the mean, generalized      218
Least square approximation      354 350
Least squares, method of      108
Least upper bound      149
Lebesgue H.      334
Lebesgue integral      185
Leibniz's rule      409
Leibniz's theorem on alternating series      245 247 248 276
Limit inferior and superior: definition of      233
Limit inferior and superior: properties of      234
Limit point: of a point set      10
Limit point: of a sequence      233
Limit tests: for convergence and divergence of improper integrals      273 281—283 285
Limit tests: for convergence and divergence of infinite series      249 250

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