×

Tensor calculus. (Le calcul tensoriel.) (French) Zbl 0174.24801

”Que sais-je?” No. 1336. Paris: Presses Universitaires de France. 127 p. (1969).
The book consists of two parts: the first represents a basic course of tensor algebra and the second gives an introduction into tensor analysis.
At the beginning, the tensor product of a finite number of finite dimensional vector spaces over the same field is defined in a very instructive way. Then the tensor algebra over a vector space is studied in both invariant and coordinate forms. After that, the exterior algebra is treated in detail and the pseudotensors are introduced. The algebraic part concludes with the study of the Euclidean tensors, i.e. the tensor algebra over a vector space with a fundamental quadratic nondegenerate form. The first part of the book is written in a successful way and familiarizes the reader with the invariant methods as well as with the classical calculus with indices.
The tensor analysis is treated firstly in the curvilinear coordinates in a Euclidean space. Then a (trivial) differentiable manifold \(V\) (i.e. \(V\) is supposed to admit global coordinates) is introduced and a Riemannian geometry is given: the absolute differential, geodesics, development of a curve, curvature tensor, Bianchi’s identities. This material is well selected and well organized. Unfortunately, the so called “classical language” (un point infiniment voisin, la partie principale, les symboles de différentiation échangeables) is used throughout the second part, which is not accordant to the present standard in this branch of mathematics.

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
53A45 Differential geometric aspects in vector and tensor analysis
15A72 Vector and tensor algebra, theory of invariants
15A75 Exterior algebra, Grassmann algebras