Tensors and Manifolds: with Applications to Mechanics and Relativity

Tensors and Manifolds [Wasserman]......Page Untitled-1 copy_0001.djvu
Title......Page p0003.djvu
Preface......Page p0004.djvu
1.1 Definitions, properties, and examples......Page p0006.djvu
1.2 Representation of vector spaces......Page p0010.djvu
1.3 Linear mappings......Page p0011.djvu
1.4 Representation of linear mappings......Page p0013.djvu
2.1 Vector spaces of linear mappings......Page p0016.djvu
2.2 Vector spaces of multilinear mappings......Page p0021.djvu
2.3 Nondegenerate bilinear functions......Page p0025.djvu
2.4 Orthogonal complements and the transpose of a linear mapping......Page p0026.djvu
3.1 The tensor product of two finite-dimensional vector spaces......Page p0031.djvu
3.2 Generalizations, isomorphisms, and a characterization......Page p0034.djvu
3.3 Tensor products of infinite-dimensional vector spaces......Page p0037.djvu
4.1 Definitions and alternative interpretations......Page p0040.djvu
4.2 The components of tensors......Page p0042.djvu
4.3 Mappings of the spaces Vsr......Page p0046.djvu
5.1 Symmetry and skew-symmetry......Page p0053.djvu
5.2 The symmetric subspace of Vºs......Page p0055.djvu
5.3 The skew-symmetric (alternating) subspace of Vºs......Page p0063.djvu
5.4 Some special properties of S²(V*) and ^²(V*)......Page p0067.djvu
6.1 Tensor algebra......Page p0079.djvu
6.2 Definition and properties of the exterior product......Page p0080.djvu
6.3 Some more properties of the exterior product......Page p0085.djvu
7.1 Maps of real cartesian spaces......Page p0092.djvu
7.2 The tangent space and the cotangent space at a point of Rⁿ......Page p0096.djvu
7.3 The tangent map......Page p0105.djvu
8.1 Definitions, properties, and examples......Page p0111.djvu
8.2 Continuous mappings......Page p0115.djvu
9.1 Definitions and examples......Page p0117.djvu
9.2 Mappings of manifolds......Page p0126.djvu
9.3 The tangent and cotangent spaces at a point of M......Page p0129.djvu
9.4 Some properties of mappings......Page p0134.djvu
10.1 Parametrized submanifolds......Page p0141.djvu
10.2 Differentiable varieties as submanifolds......Page p0143.djvu
11.1 Vector fields......Page p0146.djvu
11.2 1-form fields......Page p0154.djvu
11.3 Tensor fields......Page p0157.djvu
11.4 Mappings of tensor fields......Page p0159.djvu
12.1 Exterior differentiation of differential forms......Page p0163.djvu
12.2 Integration of differential forms......Page p0169.djvu
13.1 Integral curves and the flow of a vector field......Page p0178.djvu
13.2 Flow boxes (local flows) and complete vector fields......Page p0182.djvu
13.3 Coordinate vector fields......Page p0186.djvu
13.4 The Lie derivative......Page p0187.djvu
14.1 Completely integrable distributions......Page p0196.djvu
14.2 Completely integrable Pfaffian systems......Page p0201.djvu
14.3 The characteristic distribution of a differential system......Page p0203.djvu
15.1 Pseudo-Riemannian manifolds......Page p0209.djvu
15.2 Length and distance......Page p0215.djvu
15.3 Flat spaces......Page p0218.djvu
16.1 The Levi-Civita connection and its covariant derivative......Page p0223.djvu
16.2 Geodesics of the Levi-Civita connection......Page p0228.djvu
16.3 The torsion and curvature of a (linear or affine) connection......Page p0231.djvu
16.4 The exponential map and normal coordinates......Page p0237.djvu
16.5 Connections on pseudo-Riemannian manifolds......Page p0239.djvu
17.1 Connections between tangent spaces......Page p0243.djvu
17.2 Coordinate-free description of a connection......Page p0244.djvu
17.3 The torsion and curvature of a connection......Page p0249.djvu
17.4 Some geometry of submanifolds......Page p0255.djvu
18.1 Sympletic forms, sympletic mappings, Hamiltonian vector fields, and Poisson brackets......Page p0262.djvu
18.2 The Darboux theorem, and the natural symplectic structure of T*M......Page p0267.djvu
18.3 Hamilton\'s equations. Examples of mechanical systems......Page p0273.djvu
18.4 The Legendre transformation and Lagrangian vector fields......Page p0279.djvu
19.1 The configuration space as a pseudo-Riemannian manifold......Page p0284.djvu
19.2 The momentum mapping and Noether\'s theorem......Page p0287.djvu
19.3 Hamilton-Jacobi theory......Page p0290.djvu
20.1 Newton\'s mechanics and Maxwell\'s electromagnetic theory......Page p0299.djvu
20.2 Frames of reference generalized......Page p0304.djvu
20.3 The Lorentz transformations......Page p0307.djvu
20.4 Some properties and forms of the Lorentz transformations......Page p0312.djvu
20.5 Minkowski spacetime......Page p0316.djvu
21.1 Time dilation and the Lorentz-Fitzgerald contraction......Page p0324.djvu
21.2 Particle dynamics on Minkowski spacetime......Page p0331.djvu
21.3 Electromagnetism on Minkowski spacetime......Page p0335.djvu
21.4 Perfect fluids on Minkowski spacetime......Page p0341.djvu
22.1 Gravity, acceleration, and geodesics......Page p0346.djvu
22.2 Gravity is a manifestation of curvature......Page p0348.djvu
22.3 The field equation in empty space......Page p0351.djvu
22.4 Einstein\'s field equation......Page p0355.djvu
23.1 Schwarzschild\'s exterior solution......Page p0360.djvu
23.2 Two applications of Schwarzschild\'s solution......Page p0366.djvu
23.3 The Kruskal extension of Schwarzschild spacetime......Page p0369.djvu
23.4 The field of a rotating star......Page p0374.djvu
24.1 Schwarzschild\'s interior solution......Page p0378.djvu
2402 The form of the Friedmann-Robertson-Walker metric tensor and its properties......Page p0383.djvu
24.3 Friedmann-Robertson-Walker spacetimes......Page p0388.djvu
References......Page p0393.djvu