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| Авторизация |
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| Goursat E. — A course in mathematical analysis. Volume 2, part 2: Differential equations |
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| Предметный указатель |
Abel's theorem, applied to Euler's equation 28 15
Abelian integral 18 11
Abell 28 15
Adjoint equation, polynomial 115 42; 42
Adjoint system of equations 156 166 62
Algebraic critical points 178 63; 67; 71; 71
Algebraic differential equations 180 66; 67
Analytic extension of integrals 101 37; 56; 182 67; 68
Analytic integrals 45 22; 22; 23; 24; 25; 25; 26; 26; 26; 29; 37; 64; 84; 94
Anharmonic ratio of integrals 18 7
Antomari 246
Appell 41; 115 42;
Approximate integration of differential equations 64
Arbitrary constants 74 31 see
Asymptotic lines 43 ex. 35; ex.6
Auxiliary equation, polynomial 12 6; 43; 45; 60
Auxiliary equation, polynomial, for a system of equations 158 58; 60
Auxiliary equation, polynomial, roots of 119 43
bernoulli 11 6
Bernoulli's equation 11 6
Bertrand 41; 232 80; ex.
Bertrand's method 282 80; ex.
bessel 126 46; 62; ex.
Bessel's equation 126 46; 52; ex.8
Boole 212 ex.
Bounitzky 44 ex.
Bracket [u, v] 234 81; 83
Briot and Bouquet 45 21; 26; 64; 64; 64; 64; 65;
Briot and Bouquet's equation 178 64
Briot and Bouquet's method, analytic integrals 50; 59 26
Briot and Bouquet's theorem 175 64; 64; 64; 65
Calculus of limits 45 21 50
Calculus of limits, (system of equations) 48
Calculus of limits, (system of) 56 26; 94; 94
Calculus of limits, equations of the nth order 49 22; 37
Calculus of limits, non-linear equations 174 64
Calculus of limits, partial differential equations 58 25
Calculus of limits, system of linear equations 50 23
Calculus of limits, total differential equation 51 24
Canonical form, of a system of linear equations 161 59; 179 65
Canonical form, of substitutions 181 48; 48
Cauchy 85 18; 21; 22; 27; 30; 30; 30; 39; 46; 63; 67; 71; 71; 75; 75; 84; 86; 85; 85; Note; 86; Note; 87; 87
Cauchy — Lipschitz method 61 27; 30; 30
Cauchy's equation 257 ex.
Cauchy's first proof 68 30; 30
Cauchy's method non-homogeneous linear equations 108 39;
Cauchy's method non-homogeneous linear equations, (extended) 261 87
Cauchy's method non-homogeneous linear equations, (system of) 154
Cauchy's method non-homogeneous linear equations, partial differential equations 249 85; Note; 86; Note
Cauchy's problem 246 84; 87
Cauchy's theorem 45 22; 68; 67; 71; 71
Cauchy's theorem, (system of equations) 48 217 76
Cauchy's theorem, partial differential equations 54 26
Center, of integral curves 180 66
Center, of similitude 8 3
Characteristic curves 219 76; 77; 86; 86; 86; 87
Characteristic curves, Cauchy's method 249 257 Note; 86; Note; 87
Characteristic curves, congruence of 219 76; 76; 77
Characteristic curves, derivation from complete integral 259 86
Characteristic curves, differential equations of 219 76; 77; 77; 86 see
Characteristic developable surface 252 85; 86; Note
Characteristic direction 250 85
Characteristic equation 180 189 50; 61; 63; 54; 61; 775 66
Characteristic equation, elementary divisors 132
Characteristic equation, roots of 130 47; 48; 60; Note
Characteristic exponents 147 64; 66
Characteristic numbers 147 54
Characteristic strip 252 86; 86; Note; 87
Characteristic strip, differential equation of 262 87
Circles, differential equation of 5 1;
Circles, differential equation of, of double contact with a conic 206 ex.4
Cissoid 206 ex.
Clairaut 17 41; ex. 72; 74; ex.
Clairaut's equation 17 41; ex. 72
Clairaut's equation, generalized 212 74; ex.
Clebsch 267 88
Complete integral 236 82; 241 83; 84; Note; 91; 92
Complete integral of involutory systems 277 91 see
Complete integral, generalization of theory 272 90
Complete integral, geometric interpretation 288
Complete systems 267 88
Complete systems, change of variables 267 89
Complete systems, equivalent 268 89
Complete systems, Jacobian systems 269 89; 89; ex.; 91; 92
Complete systems, method of integration 270 89
Completely integrable total differential equations 52 24; 78
Completely integrable total differential equations, system of equations 58 24;
Complex of curves 259 86
Condition for incompressibility of a fluid 86 33
Condition for integrability of total differential equations 52 24; 78; 80
Condition for integrability of total differential equations, invariance of 281 80
Condition for integrability of total differential equations, involutory systems, Poisson's parenthesis 274 91
Condition for integrability of total differential equations, the bracket [u, v] 284 81; 83
Condition for integrability of total differential equations, the parenthesis (u, v) 234 81
Conformal representation 22 13
Congruence of curves 209 74; 76; 77
Congruence of curves, focal points of, focal surface 209 74; 77 see
Conical point 257 85
Conics, differential equation of 5 1
Conics, differential equation of, having circles of double contact 206 ex.
Conoids 220 ex.
Constant coefficients in differential equations 117 43
Constant coefficients in differential equations, (system of equations) 157 68; 68
Constant coefficients in differential equations, D'Alembert's method 122 44; 68
Constants of integration 74 31;
Continuous one-parameter groups 87 34;
Corresponding homogeneous linear equation 107 39
Cotton 64
Covariant 80 Note
Cremona transformation 198
Critical points, algebraic 178 63; 67; 71; 71
Critical points, algebraic, infinite number of 185
Critical points, algebraic, linear equations 129 47
Critical points, algebraic, non-linear equations 173 63
Critical points, algebraic, permutation of integrals about 129 47; 49;
Critical points, algebraic, transcendental 197 70
Curves, asymptotic lines 48 ex. 36; ex.
Curves, asymptotic lines, circles see Circles; cissoid 206 ex.
Curves, asymptotic lines, complex of 259 86
Curves, asymptotic lines, congruence of see "Congruence of curves"
Curves, asymptotic lines, conics see "Conics"
Curves, asymptotic lines, cycloid 41 20
Curves, asymptotic lines, edge of regression 209 74; 74; ex.
Curves, asymptotic lines, elastic space curve 99 ex.
Curves, asymptotic lines, ellipse 75 10
Curves, asymptotic lines, envelope see "Envelope"
Curves, asymptotic lines, family of 8 1
Curves, asymptotic lines, helices 220 ex.
Curves, asymptotic lines, isothermal 45 ex.
Curves, asymptotic lines, orthogonal 14 7; 17; 78;
Curves, asymptotic lines, parabola 6 1
Curves, asymptotic lines, parallel 42 ex.
Curves, asymptotic lines, similar 8 3
Curves, asymptotic lines, straight lines 4
Curves, asymptotic lines, trajectories 14 7; 17; 36
Curves, asymptotic lines, unicursal quartic 19 ex. 72; "Lines "Locus"
Cusps of integral curves 41 20; 71; 71; 73; 74; ex. see
D'Alembert 122 44; 68
D'Alembert's method 122 44; 68
Darboux 29 16; 21; ex. 85
Darboux's theorems 29 16
Definite integrals as solutions, of Bessel's equation 126 46; ex.
Definite integrals as solutions, of Laplace's equation 124 46
Delassus 286 94
Depression of order 36 19; 40
Derivative in non-linear equations, indeterminate 178 64; 65;
Derivative in non-linear equations, infinite 172 63
Developable surfaces 240 82; 85; ex. also
Differential equations 3 1
Differential equations of characteristic curves 219 76; 77; 77; 85
| Differential equations of characteristic strip 262 87
Differential equations of circles 5 1
Differential equations of conics (Halphen's method) 5 1
Differential equations of first order 6 2; 66
Differential equations of higher order 85 18; 70
Differential equations of incompressible fluid 84 33
Differential equations of isothermal curves 48 ex.
Differential equations of orthogonal trajectories 14 7; 17; 78
Differential equations of parabolas 6 1
Differential equations of similar curves 8 3
Differential equations of straight lines 4 1
Differential equations of the nth order 4 1; 2; 22; 37
Differential equations of trajectories see "Trajectories" "Special
Differential equations with periodic coefficients see "Periodic coefficients"
Differential equations, admitting a group of transformations 89 35; 35; 36; 36; 36; ex.
Differential equations, algebraic 180 66; 67
Differential equations, algebraic, of deficiency zero or one 18 11
Differential equations, Bernoulli's 11 6
Differential equations, Bessel's 126 46; 52; ex.
Differential equations, Cauchy's 257 ex.
Differential equations, Clairaut's see "Clairaut's equation"
Differential equations, Darboux's theorems 29
Differential equations, depression of order of 36 19; 40
Differential equations, Differential equations, Briot and Bouquet's 173
Differential equations, differential notation 7 2
Differential equations, elastic space curve 99 ex.
Differential equations, equations F(x,y')— 0, F(y,y') = 0 18 11
Differential equations, Euler's see Euler's equation; Euler's linear 128 45
Differential equations, existence theorems see "Existence theorems"
Differential equations, Gauss's 140 51
Differential equations, geometric representation of 14 8
Differential equations, homogeneous 8 16; 19; 35
Differential equations, integrals of see "Integral curves" "Integral
Differential equations, Jacobi's see "Jacobi's equation"
Differential equations, Lagrange's 16 9; 72; 72
Differential equations, Lamp's 146 53
Differential equations, Laplace's linear 124 46
Differential equations, Legendre's 112 ex.
Differential equations, linear 9 4; 36
Differential equations, Liouville's 79 3
Differential equations, order of 4 1
Differential equations, Painlev^'s 196 70; 70
Differential equations, Picard's 148 53
Differential equations, raising order of 41 Note
Differential equations, regular 184 50
Differential equations, Riccati's see "Riccati's equation"
Differential equations, singular points of see "Singular points"
Differential notation 7 2
Differential operators 97 36; 38; 41
Differential operators, bracket [u, v] 284 81; 83
Differential operators, Poisson's parenthesis 274 91
Differential operators, the parenthesis (u, v) 284 61
Differential operators, X[Y(f)]-Y[X(f)] 97 36; 88; 92
Dixon 44 ex.
Dominant functions 45 47 22; 23; 55 188 60; 64
Doubly periodic functions of the second kind 145 68
Edge of regression 209 74; 74; ex 2; 86
Elastic space curve 55 ex 7
Element 86; 261 87
Elementary divisors 182
Elimination, of arbitrary functions 222 77; 82; 86; 90; 93
Elimination, of constants 3 1; 74; 82; 90
Ellipsoid, lines of curvature of 41 Note
Elliptic functions 28 14
Elliptic functions as coefficients of a linear equation 144 63; 64
Elliptic functions as integrals 19 ex 3; 20; 63;
Elliptic functions existence proof from Euler's equation 28 14; 69
Elliptic functions Picard's equation 144 63
Envelope, of asymptotic lines 206 ex 6
Envelope, of integral curves 17 10; 71; 72; 74; ex 8
Envelope, of integral surfaces 288 82; 93
Envelope, of straight lines 18 10
Equations of first order, higher order see "Differential equations and special classes of equations"
Equivalent complete systems 268 89
Essentially singular points 181 47; 49
Essentially singular points, movable 196 70
Euler 19 12; 14; 14; 15; 16; ex 17; 43; 45; 69; 72; ex 3
Euler's equation 28 28 15; 69; 72
Euler's equation, Abel's theorem 28 15
Euler's equation, existence of elliptic functions 28 14; 69
Euler's equation, Lagrange's integral of 48 ex.
Euler's equation, Stieltjes's general integral 27 14
Euler's linear equation 128 45
Euler's relation for homogeneity 221 ex 3
Exceptional initial values 172 63; 64; 65
Existence theorems 45 22; ex.
Existence theorems for elliptic functions 28 14; 69
Existence theorems for integrating factors 57 26
Existence theorems for systems of partial differential equations in normal form 283 94;
Existence theorems, analytic integrals see "Analytic integrals" and "Briot and Bouquet's method"
Existence theorems, calculus of limitss see "Calculus of limits"
Existence theorems, successive approximations see "Successive approximations"
Extended group 94 36
First integrals 74 31; 31; 32; 32; 67; lb
Fixed singular points 181 66; 67
Floquet 151
Focal point 209 74
Focal surface 209 74; 77
Focus 180 65
Fuchs 134 50;
Fuchs' theorem 134 50
Functions defined by differential equations 182 67
Fundamental characteristic equation 189 50;
Fundamental system for a system of linear equations 158 56
Fundamental system of integrals 103 38; 38; 47; 47; 54
Gauss 140 51
Gauss's equation 140 51
General integral 3 1; 59 26; 31
General integral of a system of equations 57 26; 56
General integral of homogeneous linear equations 108 38; 38
General integral of partial differential equations 217 16; 82; 90
Goursat 88 170 ex. 15;
Group, differential equations admitting a 89 35; 35; 36; 36; 36; ex.
Group, differential equations of a 88
Groups of infinitesimal transformations 94 36; 36
Groups of translations 89 34;
Groups, application to differential equations 89 35; 36;
Groups, functions admitting 96 36
Groups, invariants 96 36
Groups, one-parameter continuous 86 34; 36
Groups, similar 15 34
Halphen 5 1; 42
Hedrick 255
Helices 220 ex. 83
Helicoid 220 ex. 83
Helicoid lines of, curvature of 91 36
hermite 99 ex. 63; ex.
Hoiiel 212 ex.
Homogeneity of functions, Euler's relation 221 ex.
Homogeneous equations 8 3; 19; 36
Homogeneous linear equations 102 38;
Homogeneous linear equations, (D'Alembert's method) 122 44
Homogeneous linear equations, adjoint equation, polynomial 116 42
Homogeneous linear equations, analogies with algebraic equations 113 41
Homogeneous linear equations, analogies with the Galois theory, with symmetric functions of roots 445 41
Homogeneous linear equations, auxiliary equation, polynomial 117 43
Homogeneous linear equations, Bessel's equation 126 46; 62; ex.
Homogeneous linear equations, common integrals of two equations 444 41
Homogeneous linear equations, constant coefficients 117 43
Homogeneous linear equations, corresponding 107 39
Homogeneous linear equations, critical points 129 47
Homogeneous linear equations, depression of order 400 40
Homogeneous linear equations, elliptic coefficients 444 53; 54
Homogeneous linear equations, Euchs* theorem 134 60
Homogeneous linear equations, Euler's linear equation 123 45
Homogeneous linear equations, fundamental system of integrals 103 38; 38; 47
Homogeneous linear equations, Gauss's equation 140 51
Homogeneous linear equations, general integral 103 38; 38
Homogeneous linear equations, greatest common divisor 113 41
Homogeneous linear equations, group of substitutions 132 48; 48
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