This undergraduate textbook provides a simple, concise introduction to tensor algebra and analysis, as well as special and general relativity. With a plethora of examples, explanations, and exercises, it forms a well-rounded didactic text that will be useful for any related course.
The book is divided into three main parts, all based on lecture notes that have been refined for classroom teaching over the past two decades. Part I provides students with a comprehensive overview of tensors. Part II links the very introductory first part and the relatively advanced third part, demonstrating the important intermediate-level applications of tensor analysis. Part III contains an extended discussion of general relativity, and includes material useful for students interested primarily in quantum field theory and quantum gravity.
Tailored to the undergraduate, this textbook offers explanations of technical material not easily found or detailed elsewhere, including an understandable description of Riemann normal coordinates and conformal transformations. Future theoretical and experimental physicists, as well as mathematicians, will thus find it a wonderful first read on the subject.
Author(s): Ilya Shapiro
Series: Undergraduate Lecture Notes in Physics
Publisher: Springer
Year: 2019
Language: English
Pages: 331
Front Matter ....Pages i-xviii
Front Matter ....Pages 1-1
Linear Spaces, Vectors, and Tensors (Ilya L. Shapiro)....Pages 3-20
Operations over Tensors, Metric Tensor (Ilya L. Shapiro)....Pages 21-26
Symmetric, Skew(Anti) Symmetric Tensors, and Determinants (Ilya L. Shapiro)....Pages 27-44
Curvilinear Coordinates, Local Coordinate Transformations (Ilya L. Shapiro)....Pages 45-54
Derivatives of Tensors, Covariant Derivatives (Ilya L. Shapiro)....Pages 55-65
Grad, div, rot, and Relations Between Them (Ilya L. Shapiro)....Pages 67-70
Grad, div, rot, and \(\Delta \) in Cylindric and Spherical Coordinates (Ilya L. Shapiro)....Pages 71-80
Curvilinear, Surface, and D-Dimensional Integrals (Ilya L. Shapiro)....Pages 81-99
Theorems of Green, Stokes, and Gauss (Ilya L. Shapiro)....Pages 101-108
Solutions to the Exercises from Part 1 (Ilya L. Shapiro)....Pages 109-127
Front Matter ....Pages 129-129
Maxwell Equations and Lorentz Transformations (Ilya L. Shapiro)....Pages 131-142
Laws of Relativistic Mechanics (Ilya L. Shapiro)....Pages 143-158
Maxwell Equations in Relativistic Form (Ilya L. Shapiro)....Pages 159-187
Front Matter ....Pages 189-190
Equivalence Principle, Covariance, and Curvature Tensor (Ilya L. Shapiro)....Pages 191-223
Einstein Equations, Schwarzschild Solution, and Gravitational Waves (Ilya L. Shapiro)....Pages 225-269
Basic Elements of Cosmology (Ilya L. Shapiro)....Pages 271-292
Special Sections (Ilya L. Shapiro)....Pages 293-319
Back Matter ....Pages 321-324
