1 Geometry in Regions of a Space. Basic Concepts. - 1. Co-ordinate systems. - 2. Euclidean space. - 3. Riemannian and pseudo-Riemannian spaces. - 4. The simplest groups of transformations of Euclidean space. - 5. The SerretFrenet formulae. - 6. Pseudo-Euclidean spaces. - 2 The Theory of Surfaces. - 7. Geometry on a surface in space. - 8. The second fundamental form. - 9. The metric on the sphere. - 10. Space-like surfaces in pseudo-Euclidean space. - 11. The language of complex numbers in geometry. - 12. Analytic functions. - 13. The conformal form of the metric on a surface. - 14. Transformation groups as surfaces in N-dimensional space. - 15. Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions. - 3 Tensors: The Algebraic Theory. - 16. Examples of tensors. - 17. The general definition of a tensor. - 18. Tensors of type (0 k). - 19. Tensors in Riemannian and pseudo-Riemannian spaces. - 20. The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space. Examples of invariant tensors. - 21. Rank 2 tensors in pseudo-Euclidean space and their eigenvalues. - 22. The behaviour of tensors under mappings. - 23. Vector fields. - 24. Lie algebras. - 4 The Differential Calculus of Tensors. - 25. The differential calculus of skew-symmetric tensors. - 26. Skew-symmetric tensors and the theory of integration. - 27. Differential forms on complex spaces. - 28. Covariant differentiation. - 29. Covariant differentiation and the metric. - 30. The curvature tensor. - 5 The Elements of the Calculus of Variations. - 31. One-dimensional variational problems. - 32. Conservation laws. - 33. Hamiltonian formalism. - 34. The geometrical theory of phase space. - 35. Lagrange surfaces. - 36. The second variation for the equation of the geodesics. - 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants. - 37. The simplest higher-dimensional variational problems. - 38. Examples of Lagrangians. - 39. The simplest concepts of the general theory of relativity. - 40. The spinor representations of the groups SO(3) and O(3 1). Dirac's equation and its properties. - 41. Covariant differentiation of fields with arbitrary symmetry. - 42. Examples of gauge-invariant functionals. Maxwell's equations and the YangMills equation. Functionals with identically zero variational derivative (characteristic classes). Language: English