Shapiro I.L. Lecture Notes on Vector and Tensor Algebra and Analysis

Federal University of Juiz de Fora, Brazil. — 89 p.Since the lectures have some original didactic elements, we decided to publish them. These lectures are designed for the second-third year undergraduate student and are supposed to help in learning such disciplines of the course of Physics as Classical Mechanics, Electrodynamics, Special and General Relativity. One of my purposes was, e.g., to make derivation of grad, div and rot in the curvilinear coordinates understandable for the student, and this seems to be useful for some of the students of Physics, Mathematics or Engineering. Of course, those students which are going to make career in Mathematics or Theoretical Physics, may and should continue their education using serious books on Differential Geometry like [1]. These notes are nothing but a simple introduction for beginners.Preliminary observations and notations
Vectors and Tensors
Operations over tensors, metric tensor
Symmetric, skew(anti) symmetric tensors and determinants
Curvilinear coordinates (local coordinate transformations)
Derivatives of tensors, covariant derivatives
Grad, div, rot and relations between them
Grad, div, rot and ∆ in polar, cylindric and spherical coordinates
Integrals over D-dimensional space. Curvilinear, surface and volume integrals
Theorems of Green, Stokes and Gauss