# Tensor Analysis This is the index page for self study note of tensor analysis. Though not used very often, it is worth to summaries typical problems and methods. The reference book applied is [[Introduction to Tensor Analysis and the Calculus of Moving Surfaces]]. Part I, chapters 5, 6, 7, 8, 9 are of primary focus. Besides notes of the book, some quick general notes will also be presented here. The main focus and target is on three dimensional Euclidean space (although a lot of discussion could be made on higher order of dimensions). There are possibility that it would be extended to the four dimensional Minkowski space, and very little possibility to cover the four dimensional Riemannian manifold (although appears in discussions sometimes). 1. [[Recap on coordinate systems]] 2. [[Introduction to tensor notations]] - [[Introduction to tensor notations#Einstein notation]] 3. [[Coordinate change and Jacobian]] - [[Coordinate change and Jacobian#Kronecker symbol]] 4. Tensor description of **Euclidean spaces**, an example - [[Covariant and contravariant basis, matric tensors]] - [[Fundamental elements in coordinates]] - [[The Christoffel symbol]] 5. [[General idea on tensor properties]] 6. [[The fundamental properties of tensors]] 7. [[Index juggling]] 8. [[Tensor calculus and linear algebra]] 9. [[Introduction to covariant differentiation]] 10. [[Properties of covariant differentiation]] - [[Properties of covariant differentiation#The metrinilic property|The metrinilic property]] - [[Properties of covariant differentiation#Case study A particle moving along a trajectory|Case study A particle moving along a trajectory]] 11. [[Permutation symbols and determinants]] 12. [[Relative tensor and Levi-Civita symbol]] 13. [[Cross product, curl, and higher-dimensional generalization]] General notes: - [[Basic operations]] - [[Linear algebra recap]] >[!Note] >The term "**tensor**" has its origins in Latin. It comes from the word _tensus_, which is the past participle of _tendere_, meaning "to stretch." The suffix _-or_ indicates an agent or something that performs an action, so _tensor_ originally conveyed the idea of something that stretches or extends. >The word was introduced into mathematical terminology in the 19th century by the German mathematician Woldemar Voigt in the context of elasticity theory. He used it to describe certain types of quantities that generalized the concept of scalars and vectors, emphasizing their role in "stretching" relationships across multiple dimensions. The linguistic roots reflect this "stretching" or "tension" aspect that is inherent to the concept of a tensor as it generalizes transformations across coordinate systems in physics and engineering. > >And as the origin "to stretch" suggested, performing transformation will not change the topology of the space.