The Absolute Differential Calculus: Calculus of Tensors (Dover Books on Mathematics)

A chief requirement in the study of relativity is knowledge of the absolute differential calculus, the subject that Einstein found necessary for developing his ideas mathematically. Tullio Levi-Civita was one of the founders of this field of mathematics, and he presents a clear, detailed exposition of the subject in this classic book.
The first section of the three-part treatment examines functional determinants and matrices; systems of total differential equations; linear partial differential equations in complete systems; and algebraic foundations of the absolute differential calculus, concluding with a geometrical introduction to the theory of differential quadratic forms. Part two, a study of the fundamental quadratic form and the absolute differential calculus, focuses on covariant differentiation, invariants and differential parameters, and locally geodesic coordinates; Riemann's symbols and properties relating to curvature, Ricci's and Einstein's symbols, and geodesic deviation; relations between two different metrics referred to the same parameters, manifolds of constant curvature; and differential quadratic forms of class zero and class one; and some applications of intrinsic geometry. The third and final section explores physical applications, including the evolution of mechanics and geometrical optics and their relation to a four-dimensional world according to Einstein; and gravitational equations and general relativity.