Book review

The notion of tensor is, in our days, fundamental both in mathematics and physics. But, unlike the notion of vector, the first is not an intuitive one, so it's not so easy to introduce a beginner in the world of tensors in a natural and not very difficult way. This is the case especially for physicists, not wishing to accept a new concept without understanding its necessity. I have behind me a book trying to carry out (successfully, I would say) this fairly difficult task. The preresquisites necessary for reading this book do not exceed the college algebra and geometry. The author starts from vectors in the intuitive, three dimensional euclidian space, and then generalizes them to obtain abstract, finite dimensional, vector spaces. After that, there is developed (again, only to the necessary level) the linear algebra involved in tensor calculus. At this stage, there are introduced the second order tensors, initially by means of examples in a Euclidian space (in particular, some examples are taken from continuum mechanics). Here, the tensors are defined taking into account the law of changement of their components (covariant or contravariant) when the coordinate base is changed. Only then there is given the definition of the tensor product of two Euclidian vector spaces and it is provided an algebraic definitions of tensors and are introduced the basic operations with tensors. The author considers then tensors in affine speoes, develops the tensor analysis, and gives a short description of tensors defined on a general, abstract, vector space, without any underlying geometric structure. The last two chapters of the book are devoted to the Riemannian geometry and applications to physics. The applications are taken from: mechanics, hydrodynamics, electromagnetism, quantum mechanics, gravitation and cosmology. The book is excellent written and it is to be highly recommenddd for undergraduate students in physics first of all, but also to students in mathematics, interested to see how some of the mathematical concepts as: geodesics, connexion, curvature, are used in physics. The book is suitable to be used as a textbook for an undergraduate course in tensor calculus. It would be fairly hard to find a better, more comprehensible and shorter introduction to tensors for beginners.

The book has an index, but only few other titles are listed after the introduction.