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Forsyth A.R. — Theory of differential equations. Part 1
Forsyth A.R. — Theory of differential equations. Part 1

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Название: Theory of differential equations. Part 1

Автор: Forsyth A.R.

Аннотация:

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1959

Количество страниц: 340

Добавлена в каталог: 28.10.2012

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Algebraical substitutions      274
Algebraical substitutions changed into differential substitutions      281
Arbitrary element in integral equivalent of single equation      146
Arbitrary element in integral equivalent of single equation of non-integrable system      316
Arbitrary element in integral equivalent of single equation used to modify non-integrable system      319
Arbitrary element in integral equivalent of single equation used to modify single equation      157
Arithmetic mean of two invariantive integers in transformation of simultaneous forms is a sufficient invariant      278
BACH      33
Baecklund      230
Bertrand      18 33 109 227
Bertrand's integration of an exact equation in three variables      18 109 226 227
Bertrand's integration of an exact equation in three variables generalised to exact equation in n variables      20—24
Biermann      299 312 316 321
Biermann's determination of minimum number of equations in equivalent of non-integrable system      312 316
Bilinear differential covariant      273 294
Bilinear differential covariant changed to algebraical covariant      274
Binet      148
Binet's simultaneous variations      148
Boole      67 299 305 307
Bour      1
Brioschi      211
Canonical form, of system of exact equations      43
Canonical form, reduction of two inexact equations in four variables to      325—327
Cauchy      86 118 183 211 294
Cauchy substitutions      59 86
Cauchy substitutions do not affect character of normal form      258 269
Cauchy substitutions used by Lie to transform conditioned expression      258 269
Cayley      81 99 115
Cayley's symbolical solution of equations subsidiary to first reduction of expression      100
Character of a normal form inferred from results of Natani and Frobenius      278
Character of a normal form unaffected by Cauchy substitution      258 269
Character of a normal form, conditions which determine the      253—255
Character of a normal form, conditions which determine the, and inferences from conditions      255—257
Character of a normal form, persistent      249
Christoffel      274
Class of an expression      290
Class of an expression, examples relating to      293
Clebsch      72 76 78 82—86 117 168 173 175 179 181 194 233 244 245 252 257 266 298 321
Clebsch's methods equations inferred from Lie's method      266
Clebsch's methods of treating Pfaff's problem      chap. viii
Clebsch's methods, abstract and general character of      85 86 181
Clebsch's methods, note on complete systems of partial differential equations in      76 78
Clebsch's methods, number of integrations required in      79
Collet      25 34
Collet's determination of integrating factor      25
Complete reduction of expression, after Gauss, to contain minimum number of terms      112—114
Complete systems of linear and homogeneous partial differential equations coextensive with a system of exact equations      68
Complete systems of linear and homogeneous partial differential equations in Clebsch's methods      76 78 209 225 229
Complete systems of linear and homogeneous partial differential equations in Frobenius' conditions for exact equations      52
Complete systems of linear and homogeneous partial differential equations, construction of solutions by general method of Mayer      72
Complete systems of linear and homogeneous partial differential equations, construction of solutions by simplest form of Mayer's method      74
Complete systems of linear and homogeneous partial differential equations, construction of solutions by successive derivation      68
Complete systems of linear and homogeneous partial differential equations, in generalisation of Bertrand's method      21—25
Complete systems of linear and homogeneous partial differential equations, Mayer's derivation of single solution of, from any integral of subsidiary system      76
Complete systems of linear and homogeneous partial differential equations, number of independent solutions of      69
Completely integrable systems of equations      chap. ii 300
Condition of consistency, effect of this condition on the first transformation      109
Condition of consistency, of first subsidiary system when the number of variables is odd      107
Condition of consistency, verification of, from Natani's method      164
Conditioned expression, normal form of, deduced from that of the transformed unconditioned expression      259—266
Conditioned expression, transformed by Cauchy substitutions into an unconditioned expression having normal form of same character      258 269 297
Conditions for expression in even number of variables      166 167 207
Conditions for expression in odd number of variables      168 169 207
Conditions given by Lie to determine character of normal form      253
Conditions given by Lie to determine character of normal form, inferences from      255
Conditions that partial differential equation of second order possess an intermediary integral      309
Conditions, for existence of integrals fewer in number than the general minimum, in form of interrupted products      123
Conditions, for existence of integrals fewer in number than the general minimum, in form of interrupted products with numerical interpretation      124—127
Condorcet      33
Consistency of subsidiary equations when number of variables is odd, condition for      107
Consistency of subsidiary equations when number of variables is odd, effect of this condition      109
Consistency of subsidiary equations when number of variables is odd, verification of, from Natani's method      164
Construction, gradual, of solution and normal form by Clebsch      198 199 208 215 223
Construction, gradual, of solution and normal form by Grassmann      141
Construction, gradual, of solution and normal form by Jacobi      120
Construction, gradual, of solution and normal form by Lie      257 268 270
Construction, gradual, of solution and normal form by Natani      151 157 160
Construction, gradual, of solution and normal form in Pfaff's reduction      112
Corollaries from cylindrical tangential transformation used by Lie      242 248
Corollaries from cylindrical tangential transformation used by Lie, derivable from Clebsch's results      244 245
Covariant, bilinear differential, associated with expression      273
Covariant, bilinear differential, multilinear, not associated with expression      274
Covariant, bilinear differential, no other similarly formed, exists for expression      274
Cylindrical tangential transformation      239
Cylindrical tangential transformation, equations of      241 242
Cylindrical tangential transformation, Lagrange's examples of      242
Cylindrical tangential transformation, Lie's corollaries from      242 248
Darboux      88 181 227 238 294
Darboux's investigations      88
Darboux's investigations, abstract of      chap xii
De Morgan      20
Deahna      51
Deahna's conditions of exactness of system of equations      51
Definition of tangential transformation      232
Derivation of system of exact equations from system of integrals      37
Derivation, of exact equation from one integral      1
Determinants, in algebraical transformation of bilinear form      276
Determinants, in differential transformation      94 101 103 106 166 169 201 214 254
Determination of character of normal form      253—255
Determination of character of normal form, inferences from conditions which lead to      255
Differential covariant, bilinear associated with expression      273 294
Differential covariant, none but bilinear so associated      274
Differential relation characteristic of tangential transformation      231
Differential relation characteristic of tangential transformation can be treated as a Pfaffian equation      233
Differential relation characteristic of tangential transformation, Mayer's establishment of relations among functions in      233—238
Diminution of number of equations in subsidiary system, when more than two are known      179
Diminution of number of equations in subsidiary system, when one integral is known      173
Diminution of number of equations in subsidiary system, when two integrals are known      176
Diminution, of number of equations in equivalent of a single expression when conditions are satisfied      165—169
Direct construction by Clebsch of equations which determine all the integrals of an expression      213
Dirksen      33
du Bois-Beymond      31 32
du Bois-Reymond's integration of exact equation      31 32
du Bois-Reymond's integration of exact equation compared with Natani's      32
Effect of integrals already determined, on the remainder      200 215
Effect, equations determining the, constructed direct      213
Effect, of vanishing determinant on result of first transformation      102 105
Engel      230 299 305 306
Engel on canonical form of two equations in four variables      326
Equations, determining the first integral of an expression, in an even number of variables, determining all the integrals      213 291
Equations, determining the first integral of an expression, in an even number of variables, some properties of the simultaneous, in Clebsch's method      215
Equations, determining the first integral of an expression, in an even number of variables, when conditioned      203—207
Equations, determining the first integral of an expression, in an even number of variables, when normal form involves an odd number of functions      224
Equations, determining the first integral of an expression, in an even number of variables, when unconditioned      200—203
Equivalence of two expressions through normal forms      252 285 289
Equivalent, integral of general partial differential equation of second order      320
Equivalent, integral of irreducible differential relation      250
Equivalent, integral, minimum number of equations in, of a non-integrable system      312—316
Equivalent, integral, of expression deduced from normal form      116 117
Equivalent, integral, of system of exact equations      38
Equivalent, integral, reduced forms      195
Equivalent, integral, relations between elements of      252
Equivalent, integral, solutions, of an exact equation      2
Euler      13 30 33 55 56 67 80 109
Euler's, integration of an exact equation      13 109
Euler's, integration of an exact equation, compared with Natani's      30
Euler's, integration of an exact equation, generalised to system of exact equations      55
Euler's, integration of an exact equation, view of inexact equations      80
Even reduction      105
Even reduction can always be applied to expression in 2n variables      105
Even reduction not unique      105
Exact equation, all solutions of a single, are equivalent      3
Exact equation, all solutions of a single, methods of integration of a single by integrating factor      14 25
Exact equation, all solutions of a single, methods of integration of a single, Bertrand's for three variables      18 109 226 227
Exact equation, all solutions of a single, methods of integration of a single, Bertrand's generalised for n variables      20
Exact equation, all solutions of a single, methods of integration of a single, comparison of Euler's and Natani's      30
Exact equation, all solutions of a single, methods of integration of a single, du Bois-Beymond's      31
Exact equation, all solutions of a single, methods of integration of a single, Euler's      13 109
Exact equation, all solutions of a single, methods of integration of a single, inference from Lie's general process      270
Exact equation, all solutions of a single, methods of integration of a single, Natani's      27
Exact equations, system of, any combination of solutions of, are coextensive with original system      67 69 71
Exact equations, system of, any combination of solutions of, integrating factors of      39
Exact equations, system of, any combination of solutions of, is a solution      38
Exact equations, system of, any combination of solutions of, multiplier of      40
Exact equations, system of, any combination of solutions of, partial differential equations satisfied by solution of      38
Exact equations, system of, methods of integration of, by solution of partial differential equations      67 71
Exact equations, system of, methods of integration of, Euler's generalised      55
Exact equations, system of, methods of integration of, Mayer's      59
Exact equations, system of, methods of integration of, Natani's      56
Exact equations, system of, methods of integration of, special method for two equations in four variables      62
Exact integral of system of equations satisfies a system of partial differential equations      302
Exact integral of system of equations used to modify the system      311
Exact integral of system of equations, conditions that it may exist      304 305
Exact integral of system of equations, what is meant by      300 301
Exact ordinary equations, note on and memoirs relating to      33
Exactness, Frobenius' form of      51—54
Exactness, necessary conditions are sufficient      7—12
Exactness, necessary conditions are sufficient, for a system      45—51
Exactness, necessary conditions of, for a single equation      4
Exactness, necessary conditions of, for a system      44
Exactness, number of necessary conditions independentt      6
Expression in 2n variables can always be subjected to an even reduction      105
Expression in 2n variables can always be subjected to an even reduction, completely reduced      113
Expression in 2n+1 variables can always be subjected to an odd reduction      111
Expression in 2n+1 variables can always be subjected to an odd reduction, completely reduced      114
Extensive equation solved      136 137
Extensive equation subsidiary, transformed      132 135
Extensive equation, representing single equation      122
Extensive equation, representing system of equations      122 329
Extensive variable, introduction of      121
Factor, form of, removeable from expression after transformation when the Pfaffian determinant does not vanish      90 99
Factor, form of, removeable from expression after transformation when the Pfaffian does vanish      101 105 203
FAIS      35
First integral of expression, if conditioned      207
First integral of expression, if normal form involve odd number of functions      224
First integral of expression, if unconditioned      203
First Method, Clebsch's, of treating Pfaff's problem      199—209
Frisiani      82
Frobenius      51 54 87 88 181 272 294 299
Frobenius' method of treatment of Pfaff's problem      chap. xi
Frobenius' method of treatment of Pfaff's problem, abstract and general character of      87 272
Frobenius' method of treatment of Pfaff's problem, connection of Darboux's method with      88
Frobenius, on conditions of exactness of system of equations      51—54
Functions, number of that can be obtained without affecting invariance of determinantal integer      282
Fundamental problems for non-integrable systems      311
Fundamental problems for non-integrable systems, two of the three yet unsolved      329 330
Gauss      81 112 251
Generalisation of Natani's process for a single Pfaffian inadequate to determine the equations in equivalent of system      322
Generalisation of Natani's process for a single Pfaffian to determine number of equations in equivalent of system      313
Generalisation of normal forms in even number of functions      197 252
Generalisation of normal forms in odd number of functions      222 253
Generality of integral equivalent, characteristic properties of      145 311
Gradual construction of solution and normal form by Clebsch      198 199 208 215 223
Gradual construction of solution and normal form by Grassmann      141
Gradual construction of solution and normal form by Jacobi      120
Gradual construction of solution and normal form by Lie      257 268 270
Gradual construction of solution and normal form by Natani      151 157 160
Gradual construction of solution and normal form in Pfaff's reduction      112
Grassmann      82 83 86 121 251 329
Grassmann's method of treatment of Pfaff's problem      chap. v
Grassmann's method of treatment of Pfaff's problem, abstract and general character of      83 84
Grassmann, on system of equations      122 329
Hamburger      85 331
Hamburger's comparison of methods of Natani and Clebsch      85
Hamilton      86 118 183
Hesse      35
Historical summary of methods of treating Pfaff's problem      chap. iii
Homogeneous tangential transformations      244
Homogeneous tangential transformations applied by Lie to Pfaff's problem      252
Homogeneous tangential transformations, equations characteristic of      245
Homogeneous tangential transformations, infinitesimal, a special case of      246
Imschenetsky      33 299 307
Imschenetsky on intermediary integral of partial differential equation of second order      307
Incompletely integrable system of equations      301
Incompletely integrable system of equations, example of, in subsidiary equations for partial of second order, together with conditions for two integrals      307—310
Incompletely integrable system of equations, modified by use of exact integrals      311
Incompletely integrable system of equations, number of exact integrals of      304
Independent solutions of a complete system of linear and homogeneous partial differential equations, number of      69
Inexact equations, views of, before Pfaff's memoir      80
Infinitesimal homogeneous transformation      246
Infinitesimal homogeneous transformation, equations characteristic of      247
Initial values of variables, used by Jacobi      82 183 183
Initial values of variables, used by Jacobi, introduction of, leads to principal integrals      118
Integers associated with determinants in transformation of bilinear quantics are invariantive      277
Integers associated with determinants in transformation of bilinear quantics are invariantive, arithmetic mean of the two      278
Integral equivalent, minimum number of equations in, of single equation      145 146
Integral equivalent, minimum number of equations in, of system of equations      312—317
Integral equivalent, of Pfaffian expression deduced from normal form      116 117
Integral equivalent, of Pfaffian expression deduced from normal form, of general partial differential equation of second order      320 321
Integral equivalent, of Pfaffian expression deduced from normal form, of irreducible differential relation      250 (note)
Integral, an exact, of system of equations, conditions for existence      304 305
Integral, an exact, of system of equations, must satisfy partial differential equations      302
Integral, an exact, of system of equations, used to modify system      311
Integral, an exact, of system of equations, what is meant by      300 301
Integrals of Pfaffian equation, effect of, on subsidiary system, when one is known      173
Integrals of Pfaffian equation, effect of, on subsidiary system, when two are known      178
Integrals of subsidiary system used to diminish the system when more than two are known      179
Integrals of subsidiary system used to diminish the system when one is known      173
Integrals of subsidiary system used to diminish the system when two are known      175
Integrals of subsidiary system used to diminish the system when two are known with alternative inferences      177
Integrating factor of single exact equation, by Collet's equations      25
Integrating factor of single exact equation, by De Morgan's partial equation      20
Integrating factor of single exact equation, determined, by linear equations      14—17
Integrating factor of single exact equation, general form of      4
Integrating factor of single exact equation, quotient of two is a solution      4 16
Integrating factors of system of equations      39
Integrating factors of system of equations, determinant of sets of, a multiplier      40
Integrating factors of system of equations, special method for determination of, when there are four variables      62—67
Integration of extensive equation      136 137
Intermediary integral, conditions that partial differential equation of second order possess an      307—310
Interpretation, numerical, of Grassmann's forms      124 125 133
Interrupted products      123
Interrupted products, conditions in form of, for existence of integrals fewer than general minimum      123—127
Interrupted products, interpretation of      124—127
Invariantive integers in transformation of bilinear quantics      277
Invariantive integers in transformation of bilinear quantics, arithmetic mean of two, a single sufficient invariant      278
Invariantive integers in transformation of bilinear quantics, persistence of, sufficient to ensure equivalence of two expressions      286 290
Invariantive persistence of character of normal form      249—251
Irreducible differential relation, integral equivalent of      250 (note)
jacobi      1 40 41 76 78 79 81—83 86 115 117 118 120 138 153 173 181 183 209
Jacobi's additions to Pfaff's theory      81 82
Jacobi's additions to Pfaff's theory, introduction of initial values of variables      82
Jacobi's additions to Pfaff's theory, simplification of Pfaff's reduction      117—120
Jacobi's additions to Pfaff's theory, simplification of Pfaff's solution of differential equation      183—187
Jacobi's additions to Pfaff's theory, theorem relative to four-termed expression      83 115 226
Joachimstahl      33
JORDAN      239 298
Jordan, on multipliers      40
Jordan, on tangential transformations      298
Kowalevski      331
kronecker      35
Lagrange      33 242
Lagrange's examples of cylindrical tangential transformation      242
laurent      35
Le Pont      59
legendre      242
Legendrian transformation is a tangential transformation      242
Lemmas of transformation in Clebsch's theory      210
Lexell      33
Lie      78 82 84 86—88 168 181 183 230 248 278 294
Lie's corollaries from tangential transformations      242—244 248
Lie's corollaries from tangential transformations, derivable from Clebsch's results      244
Lie's method of treatment of Pfaff's problem      chap. x
Lie's method of treatment of Pfaff's problem, abstract and character of      86 87
Lie's method of treatment of Pfaff's problem, connection herewith of Darboux's method      88
Lie's theorem for functions in tangential transformation      237
Lie's theorem for functions in tangential transformation, modified and simplified by Mayer      237
Linear transformations, introduced into treatment of Pfaff's problem      274
Linear transformations, introduced into treatment of Pfaff's problem, applied to bilinear form give two invariantive integers      277
Lipschitz      274
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