A Primer in Tensor Analysis and Relativity

Ilya L. Shapiro is a professor (professor Titular) at the Departamento de Fisica at Universidade Federal de Juiz de Fora, in Minas Gerais, Brazil. His teaching background ranges from the graduate and postgraduate level down to undergraduate and high school courses, giving him a broad perspective and authoritative voice on the subject of teaching physics. Shapiro was a member of the Russian Gravitational Association from 1988 to 1998. He is currently part of the American Mathematical Society, the Brazilian Physical Society, and as of 2015, the Accessory Committee of Physics and Astronomy in CNPq (Funding Agency of Brazilian Ministry of Science and Technology) since 2015. He is the author of more than 140 papers as well as a coauthor on the well-known 1992 book on quantum field theory and quantum gravity, Effective Action in Quantum Gravity.

Acknowledgements.- Preface.- Part I: Tensor Algebra and Analysis.- 1: Linear spaces, vectors and tensors.-  2: Operations over tensors, metric tensor.-  3: Symmetric, skew(anti) symmetric tensors and determinants.-  4: Curvilinear coordinates, local coordinate transformations.- 5: Derivatives of tensors, covariant derivates.- 6: Grad, div, rot and relations between them.- 7: Grad, div, rot and in cylindric and spherical coordinates.- 8: Curvilinear, surface and D-dimensional integrals.- 9: Theorems of Green, Stokes and Gauss.- 10: Solutions to the exercises from Part 1.- Part II: Elements of Electrodynamics and Special Relativity.- 11 Maxwell equations and Lorentz transformations.- 12 Laws of relativistic mechanics.- 13 Maxwell equations in relativistic form.- Part III Applications to General Relativity.- 14 Equivalence principle, covariance and curvature tensor.- 15 Einstein equations, Schwarzschild solution and gravitational waves.- 16 Basic elements of cosmology.- 17 Special sections.- Index.

"Shapiro's book is a promising player in the game of physics texts aimed at the modern student with correspondingly modern interests. ... this book has additional mathematical relevance. It is a very solid pedagogical effort, with the various physical themes all carefully worked out, as also the relevant mathematics, of course. Perhaps most importantly, there are many, many problems, and, as always, the student's mastery of this material is proportional to the number of problems worked." (Michael Berg, MAA Reviews, September 6, 2021)

“Shapiro’s book is a promising player in the game of physics texts aimed at the modern student with correspondingly modern interests. … this book has additional mathematical relevance. It is a very solid pedagogical effort, with the various physical themes all carefully worked out, as also the relevant mathematics, of course. Perhaps most importantly, there are many, many problems, and, as always, the student’s mastery of this material is proportional to the number of problems worked.” (Michael Berg, MAA Reviews, September 6, 2021)