Tensor Analysis
Kanisha Dezarn
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita,[1] it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
For example, in physics you start with a vector field, you decompose it with respect to the covariant basis, and that's how you get the contravariant coordinates. For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical there is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.
The metric tensor represents a matrix with scalar elements ( Z i j \displaystyle Z_ij or Z i j \displaystyle Z^ij ) and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa.
This means that if we take every permutation of a basis vector set and dotted them against each other, and then arrange them into a square matrix, we would have a metric tensor. The caveat here is which of the two vectors in the permutation is used for projection against the other vector, that is the distinguishing property of the covariant metric tensor in comparison with the contravariant metric tensor.
Two flavors of metric tensors exist: (1) the contravariant metric tensor ( Z i j \displaystyle Z^ij ), and (2) the covariant metric tensor ( Z i j \displaystyle Z_ij ). These two flavors of metric tensor are related by the identity:
For an orthonormal Cartesian coordinate system, the metric tensor is just the kronecker delta δ i j \displaystyle \delta _ij or δ i j \displaystyle \delta ^ij , which is just a tensor equivalent of the identity matrix, and δ i j = δ i j = δ j i \displaystyle \delta _ij=\delta ^ij=\delta _j^i .
2. The J \displaystyle \bar J matrix, representing the change from barred to unbarred coordinates. To find J \displaystyle \bar J , we take the "unbarred gradient", i.e. partial derive with respect to x i \displaystyle x^i :
In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. the polar gradient vector formula vs. the spherical gradient vector formula, etc.). In standard calculus, each coordinate system has its own specific formula, unlike tensor calculus that has only one gradient formula that is equivalent for all coordinate systems. This is made possible by an understanding of the metric tensor that tensor calculus makes use of.
I am a physics student and my background in math is probably enough for an introduction to tensors (I have already seen them in special rel class) , but I'm not asking for that.I want to study tensors rigorously as a mathematician would, and for that I'm willing to start relearning math from point zero. I know it's going to take a lot of time (probably a few years), but I have access to lectures and lecture notes from the math department and the patience to learn all the material. Thanks in advance.
You can't do anything without knowing linear algebra. Tensor algebra comes up with multilinear algebra then tensor calculus. Linear algebra isn't hard much more. Anyone can learn it in less than a week. Actually, in college, we weren't taught geometrical interpretation of linear algebra (saying from around India, not sure of Europe continent or other places). So if you understand the geometry of linear algebra than tensor course will be easy for you. Otherwise it would be much more harder to understand, cause geometry is hardly taught in tensor courses (in most of university, not too much of geometry is taught in tensor course).
As someone said in comment, "A good understanding of topology and metric spaces is also helpful". A person (anonymous physicist) told me that don't waste time on learning topology and also said that Einstein had done the whole general relativity without knowing topology.
I believe those who know topology understands differential geometry well, but it takes too much time to learn it (more than a week, a month, almost a year for some, it varies men to men). So better not to waste time on learning it if you are in physics major (learn it later).
The purpose of reading Bishop & Goldberg's Tensor Analysis is to gain a strong understanding of the mathematical concepts and techniques necessary for understanding and solving problems in physics that involve tensors. This book provides a rigorous and thorough treatment of tensor analysis, making it an essential resource for physicists.
The prerequisites for reading Bishop & Goldberg's Tensor Analysis include a strong background in calculus, linear algebra, and vector calculus. Familiarity with basic concepts of physics, such as mechanics and electromagnetism, is also recommended.
Yes, Bishop & Goldberg's Tensor Analysis provides a list of recommended resources in the preface, including books on advanced calculus, linear algebra, and differential geometry. Additionally, the authors provide a list of exercises at the end of each chapter to further practice the concepts learned.
While this book can be used for self-study, it is recommended to use it in a classroom setting with guidance from a professor or tutor. This will ensure a deeper understanding of the material and the ability to ask questions and receive feedback.
Bishop & Goldberg's Tensor Analysis is known for its rigorous and thorough treatment of the subject. It also includes many worked examples and exercises to help the reader fully grasp the concepts. Additionally, the book focuses on the application of tensor analysis in physics, making it particularly useful for physicists.
Introductory course in modern differential geometry focusing on examples, broadly aimed at students in mathematics, the sciences, and engineering. Emphasis is on rigorously presented concepts, tools and ideas rather than on proofs. Topics covered include differentiable manifolds, tangent spaces and orientability; vector and tensor fields; differential forms; integration on manifolds and Generalized Stokes' Theorem; Riemannian metrics, Riemannian connections and geodesics. Applications to configuration and phase spaces, Maxwell equations and relativity theory will be discussed.
Students with a Bachelor's degree will be assessed graduate level tuition rate for this course. However, one cannot receive graduate level credit for courses numbered below 400 at the University of Illinois.
Students currently registered in a University of Illinois Graduate Degree program will be restricted from registering in 16-week Academic Year-term NetMath courses. Matriculating UIUC Grad students will be allowed to register in Summer Session II NetMath courses.
This page has information regarding the self-paced, rolling enrollment course. If you are a UIUC student interested in taking a course during the summer, you may be interested in a Summer Session II course.
Has anybody used tensors in Mathematica? How to properly work with them?I find Mathematica not very friendly in this field, as I am defining my own functions for lowering & raising indices, multiplication and stuff like that.I was wondering if there is some good package or a secret way to use tensors more properly in Mathematica. For example, I need tensor analysis for general relativity kind of calculations.
TensoriaCalc - intended for basic calculations in general relativity, but not finished (calculates only Christoffel symbols, Riemann and Ricci tensor). Parallel working with many metrics is possible. Symbolic calculations are not supported.
NCAlgebra, for manipulating non-commuting algebraic expressions and computing non-commutative Grbner bases. It allows working with symbolic matrices and symbolic block matrices (e.g. symbolic block matrix inversion).
xAct - a package designed by researchers for large scale projects in general relativity; subpackages capable of extensive tensor manipulation (xTensor, xCoba) as well as perturbation theory in general relativity to any order (xPert). Other subpackages can also work with tensor spherical harmonics, spinor computations as well as exterior calculus (diferential forms).
OGRe (free) - released in 2021 for Mathematica 12.0 and later. Designed to be both powerful and user-friendly. Especially suitable for general relativity. Allows performing arbitrarily complicated tensor operations, and automatically transforms between index configurations and coordinate systems behind the scenes as needed for each operation.
I've started self studying tensor calculus, my sources are the video lecture series on the YouTube channel; "MathTheBeautiful" and the freeware textbook/notes; "Introduction to Tensor Calculus" by Kees Dullemond & Kasper Peeters. Other textbooks go much more in depth in advanced math topics. I have been through the first 3 chapters and watched the first 5 videos, but I don't seem to understand the content. I don't know what I should take from these lectures and notes and what part of the work to focus on in order to start practicing as soon as possible.