Tensor Calculus On Time Scales Dynamic Calculus And Riemannian Spaces (Svetlin G. Georgiev;)

[Farid-Khan]

English | 2026 | ISBN: 3112228480 | 309 pages | True PDF,EPUB | 54.49 MB

Catergory: Mathematics, Science, Nonfiction

In chapter 1 N dimensional spaces, contravariant vectors, covariant vectors and invariants are introduced. Some of their properties are deduced. Transformations of coordinates are investigated. Chapter 2 provides an informative introduction concerning the origin and nature of the tensor concept and the scope of the tensor calculus. The tensor algebra has been developed in an N dimensional space. Contravariant, covariant and mixed tensors of arbitrary order are defined. Some of their properties are deduced. The quotient law of the tensors is formulated and proved. Outer product and contractions are introduced. In Chapter 3, an N dimensional Riemannian space has been chosen for the development of tensor calculus. Metric and associated tensors are defines and some of their properties are explored. Affine and curvalinear coordinates are introduced. In Chapter 4, the Christoffel symbols of the first and second kinds are defined. Some of their basic properties are established. Covariant derivatives are introduced. The divergence, Laplace operator and curl are defined and explored. Intrinsic differentiation is studied. Chapter 5 is devoted to Riemann-Christoffel tensor, Ricci tensor, covariant curvature tensor, Riemann curvature and Einstein tensor and we deduct some of their properties. In Chapter 6, we represent some applications of tensor calculus in relativistic dynamics and relativistic kinematics. Lorentz transformations are derived on arbitrary time scales.

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