Tensor analysis: Transformation of coordinates, The summation convention, Contravariant vectors,
Invariants, Covariant vectors, Tensors, The Christofell 3-index symbols and their relations, Riemann
symbols and the Riemann tensor, The Ricci tensor, Quadratic differential forms, The equivalence of
symmetric quadratic differential forms, Covariant differentiation with respect to a tensor gij,
Introduction to a metric: Definition of a metric, N-tuply orthogonal systems of hypersurfaces in a Vn,
Metric properties of a space Vn immersed in a Vm, Geodesics, Riemannian, Normal and geodesic
coordinates, Geodesic form of the linear element, Finite equations of geodesics, Curvature of a
curve, Parallel displacement and the Riemann tensor, Fields of parallel vectors, Associate directions,
Curvature of Vn at appoint, The Bianchi identity, The theorem of Schur, Isometric correspondence of
spaces of constant curvature, Conformal spaces, Spaces conformal to flat space, Orthogonal
ennuples: The Frenet formulas Principal directions determined by a symmetric covariant tensor of
the second order, The Ricci principal tensors, Condition that a congruence of an orthogonal ennuple
be normal, N-tuply orthogonal systems of hypersurfaces, N-tuply orthogonal systems of
hypersurfaces in a space conformal to a flat space, Congruence canonical with respect to a given
congruence, Recent developments.

The student will learn some basics as well as some advanced topics of Tensor analysis.

A good and mathematical introductory course

L.P. Lebedev, Tensor Analysis, World Scientific

E. Nelson, Tensor Analysis, Princeton University Press

Albert Joseph McConnell, Applications of Tensor Analysis, Dover Publications

Mikhail Itskov, Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics, Springer