Mathematics

List of ContributorsPrefacePart I. Tensor Analysis Introduction 1. Tensor Algebra 1.1. Scope of the Section 1.2. Curvilinear Coordinates 1.3. Affine Geometry 1.4. Vector Space 1.5. Geometric Objects 1.6. Scalars, Vectors, and Tensors 1.7. Base Vectors and Reciprocal Bases 1.8. Tensor Algebra 1.9. Multivectors 1.10. Second-Order Tensors 1.11. Normal Form of Symmetric Tensors 1.12. Normal Form of a Bivector 2. Tensor Analysis 2.1. Scope of the Section 2.2. Metric Tensor 2.3. Anholonomic Components of Tensors 2.4. Physical Components of Tensors 2.5. Covariant Differentiation 2.6. Invariant Differential Operators 2.7. Intrinsic Differentiation 2.8. The Lie Derivative 2.9. The Riemann-Christoffel Tensor 3. Geometry of Subspaces 3.1. Scope of the Section 3.2. Curvilinear Coordinates for a Surface in e3 3.3. Subspace Xm of Xn 3.4. Sections and Reductions of Tensors 3.5. Total Covariant Differentiation 3.6. Curves in Space 3.7. Hypersurface Xn-1 in Rn 3.8. The Volume of Xm in Xn 3.9. Stokes' Theorem 4. Nonriemannian Geometry 4.1. Scope of the Section 4.2. Affine Connection 4.3. Geodesies 4.4. Curvature 4.5. Some Identities Involving the Curvature Tensor 4.6. Covariant Differentiation and Curvature Tensor in Anholonomic Coordinates 5. Geometry of Finsler 5.1. Scope of the Section 5.2. Finsler Spaces 5.3. Metric Tensor Derivable from a Function 5.4. Covariant Differentiation 5.5. Torsion and Curvature Tensor ReferencesPart II. Group Theory Introduction 1. Groups and Semigroups 2. Lattices and Morphisms 3. Lie Groups 4. Linear Algebras, Frobenius and Lie 5. Crystallographic Groups ReferencesPart III. Theory of Invariants 1. Introduction 1.1. Invariants of Vectors and Tensors 1.2. Reducible and Irreducible Invariants; Integrity Bases 1.3. Results from Classical Theory 1.4. The Orthogonal Groups and Certain Subgroups 2. Isotropy 2.1. Integrity Bases for Vectors 2.2. Isotropic Tensors 2.3. Invariants of Vectors and Second-Order Tensors; General Forms 2.4. Results concerning Traces of Matrix Polynomials 2.5. Invariants of Symmetric Second-Order Tensors 2.6. Invariants of Vector and Symmetric Second-Order Tensors; Proper Orthogonal Group 2.7. Full Orthogonal Group; Invariants of Vectors and Second-Order Tensors 3. Transverse Isotropy 3.1. Invariants of Vectors and Tensors; General Forms 3.2. Relations for Matrix Polynomials in 2X2 Matrices 3.3. Invariants of Symmetric Second-Order Tensors 3.4. Invariants of Symmetric Second-Order Tensors and Vectors 3.5. Syzygies for the Invariants 4. The Crystal Classes 4.1. Theorems concerning Integrity Bases 4.2. Invariants of a Symmetric Tensor 4.3. Syzygies between the Invariants of a Symmetric Tensor 4.4. Invariants of a Symmetric Tensor and a Vector 4.5. Invariants of an Arbitrary Number of Vectors 5. Tensor Polynomial Functions of Vectors and Tensors 5.1. General Statements 5.2. Examples of Tensor and Vector Polynomial Functions; Proper Orthogonal Transformation Group 5.3. Examples of Tensor and Vector Polynomial Functions; Full Orthogonal Transformation Group 6. Invariant Functionals; Vector and Tensor Functionals 6.1. General Statements 6.2. Differential Approximations to Functionals 6.3. Integral Approximations to Functionals 7. Minimality of the Integrity Bases 7.1.