# What is Tensor analysis: Definition and 76 Discussions

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.
Many mathematical structures called "tensors" are tensor fields. For example, the Riemann curvature tensor is not a tensor, as the name implies, but a tensor field: It is named after Bernhard Riemann, and associates a tensor to each point of a Riemannian manifold, which is a topological space.

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1. ### Need resources for self study of tensor analysis and other Physics topics

Hello. I'm currently trying to learn tensor analysis using MATHEMATICAL METHODS FOR PHYSICISTS(By Arfken) but I cannot understand this book well. Is there any other book (same level as Arfken) to learn about tensor analysis as a beginner? Is there anyway that I find out wchich books are...

7. ### I Excellent 3D Graphics thing at math3d.org

May I recommend math3d.org as a website for making 3D-Graphics for fresh_42's page on List of Online Calculators for Math, Physics, Earth and Other Curiosities. I also might recommend cloudconvert.com to make .gifs. It is simpler than ezgif.com, mainly because it is not so feature rich. Here...
8. ### I Differential k-form vs (0,k) tensor field

Hi, I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field. Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...

43. ### Transformation rule for product of 3rd, 2nd order tensors

1. Problem statement: Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations : [/B] Transformation rule for 3rd order tensors: Z'ijk =...
44. ### Deriving Riemann Tensor Comp. in General Frame

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
45. ### Classical I need a fluid mechanics textbook

Hello, everyone, please forgive me for my poor English. I'm a sophomore, major in Astronomy. I've finished Hassani's book(Mathematical Physics). And I've learned Real variable function and functional analysis （I do not know what exact name of this course） I'd like to buy a textbook with...
46. ### Tensor Analysis in vector and matrix algebra notation

Is there anywhere that teaches tensor analysis in both tensor and non tensor notation, because I'm having to pause each time i look at something in tensor notation and phrase it mentally in non tensor notation at which point it becomes staggeringly simpler. Any help apreciated
47. ### Studying Reading Bishop & Goldberg's Tensor Analysis: Prerequisites for Physicists

I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand...
48. ### Preliminary knowledge on tensor analysis

I am not sure whether this needs to be transported in another topic (as academic guidance). I have some preliminary knowledge on tensor analysis, which helps me being more confident with indices notation etc... Also I'm accustomed to the definition of tensors, which tells us that a tensor is an...
49. ### Where Can I Find a Comprehensive Tensor Analysis Workbook?

Would anybody have some good recommendations for a workbook on tensor analysis? I'm looking for the kind of book that would ask a ton of basic questions like: "Convert the vector field \vec{A}(x,y,z) \ = \ x^2\hat{i} \ + \ (2xz \ + \ y^3 \ + \ (xz)^4)\hat{j} \ + \ \sin(z)\hat{k} to...
50. ### Question on generalized inner product in tensor analysis

Hello, some time ago I read that if we know the metric tensor g_{ij} associated with a change of coordinates \phi, it is possible to calculate the (Euclidean?) inner product in a way that is invariant to the parametrization. Essentially the inner product was defined in terms of the metric...