The absolute differential calculus (calculus of tensors)

The Absolute Differential Calculus [Levi-Civita]......Page 13126056_0001.djvu
Title......Page 0002_0001.djvu
Preface......Page 0004.djvu
Preface to the First (Italian) Edition......Page 0006.djvu
Contents......Page 0010.djvu
1.1 Geometrical terminology......Page 0016.djvu
1.2 Functional determinants and change of variables......Page 0017.djvu
1.3 The fundamental theorem on implicit functions......Page 0018.djvu
1.4 Effect on a functional determinan t of a change of variables......Page 0019.djvu
1.5 The necessary and sufficient conditions for the independence of n functions of n variables......Page 0020.djvu
1.6 Functional matrices. Definition of the independence of m functions of n variables......Page 0023.djvu
1.7 Theorem......Page 0024.djvu
2.1 Preliminary remarks......Page 0028.djvu
2.2 Conditions necessary for integrability. Completely integrable, or complete, systems......Page 0030.djvu
2.3 The integration of a mutually consistent system can always be reduced to that of a complete system......Page 0031.djvu
2.4 Bilinear covariants and the rsulting form for the conditions of complete integrability......Page 0033.djvu
2.5 Morera's method of integration......Page 0037.djvu
2.6 Note on Mayer's method......Page 0040.djvu
2.7 Application......Page 0041.djvu
2.8 Mixed systems of equations......Page 0044.djvu
3.1 Linear operators......Page 0048.djvu
3.2 Integrals of an ordinary differential system and the partial differential equation which determines them......Page 0051.djvu
3.3 Principal integrals......Page 0053.djvu
3.4 Independent integrals. General integral......Page 0055.djvu
3.5 Direct study of the most general linear homogeneous partial differential equation......Page 0056.djvu
3.6 Integrals of a total differential system, and the associated system of partial differential equations which determines them......Page 0062.djvu
3.7 Principal integrals, as typical cases of independent integrals......Page 0063.djvu
3.8 The general integral......Page 0065.djvu
3.9 Direct study of the most general system of linear homogeneous partial differential equations of the first order. Complete systems. Jacobian systems......Page 0067.djvu
3.10 Equivalence of every complete system to a Jacobian system with the same number of equations. Note on Cramer's rule......Page 0068.djvu
3.11 Integration by means of the associated system......Page 0072.djvu
4.1 Effects on some analytical entities of a change of variables......Page 0076.djvu
4.2 m-fold systems. Forms of degree m and m-ply linear forms......Page 0080.djvu
4.3 Invariance, covariance, and contravariance of a simple system with respect to linear transformations. Dual variables......Page 0082.djvu
4.4 Invariance, covariance, and contravariance of an m-fold system with respect to linear transformations. Mixed systems or tensors. Vanishing of a tensor an invariant property......Page 0084.djvu
4.5 Symmetrical double systems......Page 0087.djvu
4.6 Sets of n covariant and contravariant simple systems. Theorem on reciprocal sets......Page 0089.djvu
4.7 Addition of tensors......Page 0090.djvu
4.8 Multiplication of tensors......Page 0091.djvu
4.9 Contraction of tensors......Page 0092.djvu
4.10 Composition of tensors......Page 0094.djvu
4.11 Change of variables in general. m-fold systems whose elements are functions of position. First general definition of a tensor. Typical tensors of rank 1......Page 0095.djvu
4.12 Second general definition of tensors whose elements are functions of position. Examples......Page 0098.djvu
4.13 More complex laws of transformation. Scope of the Absolute Differential Calculus......Page 0100.djvu
5.1 Parametric equations of a surface......Page 0101.djvu
5.2 Expressions of ds²......Page 0103.djvu
5.3 Determination of the directions drawn from a generic point......Page 0105.djvu
5.4 Angle between two directions. Contravariance of the coefficients a[ik]......Page 0107.djvu
5.5 Associated, and in particular reciprocal, tensors. The typical example of the parameters and moments of a single direction......Page 0110.djvu
5.6 Surface tensors......Page 0111.djvu
5.7 Parameters and moments of the co-ordinate lines. Element of area......Page 0113.djvu
5.8 Fundamental observation (Gauss's) on the intrinsic geometry of a surface......Page 0114.djvu
5.9 Note on developable surfaces......Page 0115.djvu
5.10 Geometrical definition......Page 0116.djvu
5.11 First consequences. Equipollence of vectors with respect to a surface......Page 0118.djvu
5.12 Infinitesimal displacement. Infinetismal form of the law of parallelism......Page 0119.djvu
5.13 The intrinsic character of parallelism......Page 0121.djvu
5.15 Intrinsic equations of parallelism......Page 0122.djvu
5.16 Christoffel's symbols......Page 0126.djvu
5.17 Equations of parallelism in terms of covariant components......Page 0127.djvu
5.18 Some analytical verifications......Page 0129.djvu
5.19 Permutability......Page 0130.djvu
5.20 n-dimensional manifolds......Page 0134.djvu
5.21 Euclidean manifolds. Any Vn can always be considered as immersed in a Euclidean space......Page 0136.djvu
5.22 Angular metric......Page 0138.djvu
5.23 Definition of geodesics......Page 0143.djvu
5.24 Differential equations of geodesics......Page 0146.djvu
5.25 Geodesic curvature......Page 0150.djvu
5.26 Extension of the notion of parallelism. Bianchi's derived vectors......Page 0152.djvu
5.27. Autoparallelism of geodesics......Page 0155.djvu
5.28 remarks on the case of an indefinite ds²......Page 0156.djvu
6.1 Covariant differentiation......Page 0159.djvu
6.2 Particular cases......Page 0162.djvu
6.3 Ricci's lemma......Page 0163.djvu
6.5 Conservation of the rules of the ordinary differential calculus......Page 0164.djvu
6.6 Applications......Page 0167.djvu
6.7 Divergence of a vector and of a double tensor. DELTA2 of an invariant......Page 0168.djvu
6.8 Some laws of transformation. ε-systems. Vector product. Extension of a field......Page 0171.djvu
6.9 Rotor of a simple tensor in three dimensions......Page 0176.djvu
6.10 Sections of a manifold. Geodesic manifolds......Page 0177.djvu
6.11 Locally geodesic (or locally Cartesian) co-ordinates......Page 0179.djvu
6.12 Severi's theorem......Page 0186.djvu
7.1 Cyclic displacement and the relations between parallelism and curvature......Page 0187.djvu
7.2 Cyclic displacement round an elementary parallelogram......Page 0188.djvu
7.3 Fundamental properties of Riemann's symbols of the second kind......Page 0192.djvu
7.4 Fundamental properties and number of Riemann's symbols of the first kind......Page 0194.djvu
7.5 Bianchi's identities......Page 0197.djvu
7.6 Commutation rule for the second covariant derivatives......Page 0199.djvu
7.7 Cyclic displacement round any infinitesimal circuit......Page 0201.djvu
7.9 Application to surfaces. Gaussian curvature of a Vn......Page 0208.djvu
7.10 Riemannian curvature of a Vn......Page 0210.djvu
7.11 Case of a V3. The tensors α[ik] of Ricci and G[ik] of Einstein......Page 0213.djvu
7.12 Curvature of a manifold of three dimensions around a point. Principal directions and invariants......Page 0216.djvu
7.13 Geodesics infinitely near a given geodesic......Page 0223.djvu
7.14 Geodesic deviation in an n-dimensional manifold......Page 0224.djvu
7.15 Invariant form of the equations defining geodesic deviation......Page 0225.djvu
7.16 Geodesic deviation. Specification of the differential system. First integral. Linear relation in finite terms......Page 0230.djvu
7.17 Reduced form of the differential system (I) in terms of the co-ordinates y......Page 0233.djvu
8.1 Differences between Christoffel's symbols relative to two different metrics assigned to the same analytical manifold......Page 0235.djvu
8.2 Differences between the covariant derivatives......Page 0237.djvu
8.3 Differences between Riemann's symbols......Page 0239.djvu
8.4 Case of two metrics in conformal representation......Page 0243.djvu
8.5 Isotropic manifolds......Page 0247.djvu
8.6 Schur's theorem......Page 0250.djvu
8.7 Canonical form of ds² for a manifold of constant curvature......Page 0251.djvu
9.1 Forms of class zero (or Euclidean forms)......Page 0257.djvu
9.2 Conformal representation of a manifold of constant curvature on a Euclidean space. Mutual applicability of all Vn's with the same constant curvature......Page 0261.djvu
9.3 General remarks on hypersurfaces in Euclidean space. Second fundamental form......Page 0264.djvu
9.4 Forms of class 1 (hypersurfaces in Euclidean space)......Page 0268.djvu
9.5 Hyperspherical representation and curvature of a hypersurface......Page 0273.djvu
10.1 general remarks on congruences. geodesic and normal congruences......Page 0276.djvu
10.2 Sets of n congruences. Determination of a vector by n invariants......Page 0280.djvu
10.3 Geometrical definitions of Ricci's coefficients of rotation......Page 0283.djvu
10.4 Commutation formula for the second derivatives along the arcs......Page 0288.djvu
10.5 Case in which one of the congruences of the set is geodesic......Page 0289.djvu
10.7 Case in which one of the congruences of the set is normal. Complete normality. Differential relations satisfied in every case by the y's......Page 0290.djvu
10.8 Canonical system with respect to a given congruence......Page 0293.djvu
10.9 Congruences of straight lines in Euclidean space. Geometrical significance of the canonical system......Page 0297.djvu
11.1 Hamilton's principle for a free particle......Page 0302.djvu
11.2 Time as a fourth co-ordinate. Spacetime. World lines......Page 0304.djvu
11.3 General transformations of co-ordinates in spacetime. Simultaneity......Page 0305.djvu
11.4 Einstein's form for Hamilton's principle. Its invariant character under any transformation of co-ordinates......Page 0306.djvu
11.5 Mass and energy: views suggested by the modification of the dynamical law......Page 0309.djvu
11.6 Einstein's form for the principle of inertia. Restricted relativity......Page 0313.djvu
11.7 The kinematics of rigid systems. Ordinary method of approach and possible variants......Page 0316.djvu
11.8 Römerian units. Study of Lorentz transformations......Page 0321.djvu
11.9 Relative motion. Composition of velocities. Kinematical justification of a formula of Fresnel's......Page 0331.djvu
11.10 Further generalization of the metric of V4, still coinciding to a first approximation with ordinary dynamics......Page 0335.djvu
11.11 An important particular case. Corresponding trajectories and their identity with those of an ordinary mechanical problem......Page 0338.djvu
11.12 Qualitative characteristics of relativity metrics. Geodesic principle for the dynamics of a material particle. Stationary and, in particular, statical line elements......Page 0340.djvu
11.13 Versors in a V4 with pseudo-Euclidean metric......Page 0344.djvu
11.14 Digression on geodesics of zero length......Page 0345.djvu
11.15 Some elementary theorems of geometrical optics......Page 0349.djvu
11.16 Geometrical optics according to Einstein and the meaning of the constant c......Page 0350.djvu
11.17 Interpretation in geometrical optics of the condition ds²=0......Page 0353.djvu
11.18 Fermat's principle in stationary relativity metrics......Page 0355.djvu
11.19 The stress tensor and its divergence in the classical theory......Page 0359.djvu
11.20 The fundamental equations of the mechanics of continuous systems, referred to fixed axes; transformations of them in general co-ordinates (space co-ordinates)......Page 0362.djvu
11.22 Equivalent form for the system (52) and (53)......Page 0364.djvu
11.23 Einsteinian modification of the equations of motion of a continuous system in a particular case......Page 0366.djvu
11.24 General case. Introduction of the energy tensor, and meaning of its components in general co-ordinates......Page 0369.djvu
11.25 Relativistic form of the equations of motion of a continuous system......Page 0374.djvu
11.26 A particular class of motions of a continuous system......Page 0375.djvu
11.27 Experimental determination of the coefficients of an Einsteinian ds²......Page 0378.djvu
12.1 Qualitative properties of the coefficients of ds²......Page 0384.djvu
12.2 The tensor G[ik] and its divergence. The gravitational tensor......Page 0386.djvu
12.3 Solidarity of physical phenomena. Criteria for the construction of the gravitational equations, and reduction of the inductive proof of their validity to the statical case......Page 0389.djvu
12.4 General equations of Einsteinian statics. Empty space......Page 0393.djvu
12.5 First approximation. Connection with Poisson's equation......Page 0398.djvu
12.6 The Einsteinian ds² which corresponds to a first approximation to an assigned Newtonian field......Page 0403.djvu
12.7 Further approximation for the coefficient g∞ = V² in statical conditions......Page 0407.djvu
12.8 A theorem of mechanical equivalence......Page 0409.djvu
12.9 Motion of the planets according to Einstein, to a second approximation. Displacement of perihelion......Page 0411.djvu
12.10 Displacement of the spectral lines. Deflection of light......Page 0415.djvu
12.11 Three-dimensional metrics with spherical symmetry......Page 0423.djvu
12.12 Digression on the calculation of curvatures......Page 0429.djvu
12.13 The gravitational equations in the case of spherical symmetry. Schwarzschild's rigorous solution......Page 0434.djvu
12.14 Spatially uniform metrics; their cosmological interest......Page 0440.djvu
12.16 De Sitter's solution......Page 0444.djvu
12.17 Einstein's additional term. Indication of other rigorous solutions......Page 0452.djvu
Name Index......Page 0456.djvu
Subject Index......Page 0458.djvu