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. MatinSAR

    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...
  2. cianfa72

    I Calculation of Lie derivative - follow up

    Hi, a doubt related to the calculation done in this old thread. $$\left(L_{\mathbf{X}} \dfrac{\partial}{\partial x^i} \right)^j = -\dfrac{\partial X^j}{\partial x^i}$$ $$L_{\mathbf{X}} {T^a}_b = {(L_{\mathbf{X}} \mathbf{T})^a}_b + {T^{i}}_b \langle L_{\mathbf{X}} \mathbf{e}^a, \mathbf{e}_i...
  3. A

    Relativity Looking for a good introductory Tensor Analysis Textbook

    Hello all, I've taken math through differential equations and linear algebra, am in my senior year of physics curricula while conducting McNair research regarding General Relativity. I found a NASA document outlining Einstein's field equations, which suggests only preparative familiarity with...
  4. SH2372 General Relativity (8): Tensor operations

    SH2372 General Relativity (8): Tensor operations

  5. SH2372 General Relativity (7): Tensor components and coordinate transformations

    SH2372 General Relativity (7): Tensor components and coordinate transformations

  6. yucheng

    Unit Basis Components of a Vector in Tensorial Expressions?

    Divergence formula $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$ If we express it in terms of the components of ##\vec{A}## in unit basis using $$A^{*j} = \sqrt{g^{jj}} A^{j}$$ , we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}}...
  7. George Keeling

    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. cianfa72

    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...
  9. S

    I Proving ##\partial^{i} = g^{ik} \partial_{k}##

    Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that $$ \frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k} $$ I wanted to prove this identity. Using the chain rule $$ \frac{\partial}{\partial x_{i}}=\frac{\partial...
  10. L

    I Physical parameters for spin 1/2 particles

    I am having trouble to understand what it means by "physically relevant real parameters" and how does it help us to specify a quantum system. Let say, we have a state of k half spin electrons? My guess is about the local phase of the spin, and this would make it 2^k parameters since each...
  11. T

    I Can Tensors of Any Rank Be Approximated by Matrices?

    Hello. Could we approximate a tensor of (p,q) rank with matrices of their elements? I am talking also about the general case of a tensor not only special cases. For example a (2,0) tensor with i, j indices is a matrix of ixj indices. A (3,0) tensor with i,j,k indices I think is k matrices with...
  12. F

    Find the tensor that carries out a transformation

    I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##. ##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}## ##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
  13. K

    Show that a (1,2)-tensor is a linear function

    I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
  14. Bright Liu

    How do I derive this vector calculus identity?

    ##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B## ##\mathcal I## is the unit tensor
  15. Haorong Wu

    Classical Books about tensor analysis just good enough for physics

    Hi. I am looking for a book about tensor analysis. I am aware that there have been some post about those books, but I wish to find a thin book rather than a tome but just good enough for physics, such as group theory, relativistic quantum mechanics, and quantum field theory. I am reading...
  16. Q

    I Metric Tensor: Symmetry & Other Constraints

    Aside from being symmetric, are there any other mathematical constraints on the metric?
  17. S

    Infinitesimal coordinate transformation of the metric

    I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha## ##\frac{\partial x^\alpha}{\partial...
  18. S

    Antisymmetry of the electromagnetic field tensor

    I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
  19. S

    Contracting one index of a metric with the inverse metric

    Since ##\nu## is contracted, we form the scalar product of the metric and inverse metric, ##g_{\mu\nu}g^{\nu\lambda} = (\vec{e_\mu} \cdot \vec{e_\nu}) \cdot (\vec{e^\nu} \cdot \vec{e^\lambda}) = \vec{e_\mu} \cdot (\vec{e_\nu} \cdot \vec{e^\nu}) \cdot \vec{e^\lambda} = \delta^\lambda_\mu## I...
  20. A

    I Contravariant Vector Transformation in Spherical Polar Coordinates

    In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$. But from the transformation of contravariant vector...
  21. V

    Gauge invariance in GR perturbation theory

    I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the metric. We use an arbitrary small vector field to have the effect of a chart transition map or...
  22. Abhishek11235

    A Finding the unit Normal to a surface using the metric tensor.

    Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
  23. BookWei

    Velocity is a vector in Newtonian mechanics

    I studied the vector analysis in Arfken and Weber's textbook : Mathematical Methods for Physicists 5th edition. In this book they give the definition of vectors in N dimensions as the following: The set of ##N## quantities ##V_{j}## is said to be the components of an N-dimensional vector ##V##...
  24. Sayak Das

    Finding the inverse metric tensor from a given line element

    Defining dS2 as gijdxidxj and given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...
  25. P

    Dust in special relativity - conservation of particle number

    Homework Statement My textbook states: Since the number of particles of dust is conserved we also have the conservation equation $$\nabla_\mu (\rho u^\mu)=0$$ Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is...
  26. saadhusayn

    I Transformation of covariant vector components

    Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$: $$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$ And $$ \textbf{e}^i = \nabla u^{i} $$ In the primed...
  27. W

    Calculus Vector Analysis and Cartesian Tensors by Bourne and Kendall

    I have to do a teaching assistant job on a multivariable calculus class, I have to survey books that can be useful as resources. Has anyone used this book by Bourne and Kendall? I noticed that the treatment of vector analysis seems good and the chapter on Cartesian tensors seem to be a good...
  28. arpon

    I Is the Contraction of a Mixed Tensor Always Symmetric?

    Is that true in general and why: $$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$
  29. M

    Vector Calculus - Tensor Identity Problem

    Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...
  30. W

    Geometry Tensors and Manifolds by Wasserman

    I would like to know at what level is the book Tensors and Manifolds by Wasserman is pitched and what are the prerequisites of this book? Given the prerequisites, at what level should it be (please give examples of books)? If anyone has used this book can you please kindly give your comments and...
  31. Diego Berdeja

    I Lorentz Transformations in the context of tensor analysis

    Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...
  32. ibkev

    B Tensor Calculus vs Tensor Analysis?

    I've seen the terms tensor calculus and tensor analysis both being used - what is the difference?
  33. Jianphys17

    Differential Geometry book with tensor calculus

    Hi, there is a book of dg of surfaces that is also about tensor calculus ? Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus! Thank you in advance
  34. BiGyElLoWhAt

    I How Do You Correctly Apply Indices in Tensor Calculus for Curvature?

    Here's what I'm watching: At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate ##D_s D_r T_m - D_r D_s T_m## And on the last term I'm having some difficulties, the second christoffel symbol. So we have ##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
  35. W

    Riemann tensor given the space/metric

    Homework Statement Given two spaces described by ##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv## ##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv## Calculate the Riemann tensor Homework Equations Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...## We have...
  36. W

    Covariant derivative of vector fields on the sphere

    Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...
  37. W

    Locally flat coordinates on the Poincaré half plane

    Homework Statement Find the locally flat coordinates on the Poincaré half plane. Problem I.6.4 by A. Zee Homework Equations [/B] Poincaré Metric: ##ds^2 = \frac{dx^2 + dy^2}{y^2}## The Attempt at a Solution First, I'm having problems with the explanation in Zee's book. He said that we can...
  38. W

    Find the coordinate transformation given the metric

    Homework Statement Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##. Homework Equations ##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy## ##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
  39. W

    Metric for the construction of Mercator map

    Homework Statement The familiar Mercator map of the world is obtained by transforming spherical coordinates θ , ϕ to coordinates x , y given by ##x = \frac{W}{2π} φ, y = -\frac{W}{2π} log (tan (\frac{Θ}{2}))## Show that ##ds^2 = Ω^2(x,y) (dx^2 + dy^2)## and find ##Ω## Homework Equations...
  40. W

    Prove the transformation is scalar

    Homework Statement 1.) Prove that the infinitesimal volume element d3x is a scalar 2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar. Homework EquationsThe Attempt at a Solution [/B] 1.) Rkh = ∂x'h/∂xk By coordinate transformation, x'h = Rkh xk dx'h =...
  41. W

    Construction of an affine tensor of rank 4

    Homework Statement In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which ∑k ∑l Cijkl Bkl = Bij + Bji ∑k ∑l Dijkl Bkl = Bij - Bji are identities. Homework Equations The Attempt at a...
  42. R

    Proving Spin Coefficient Transformation for Null Rotation with l Fixed

    In Newmann-Penrose formalism, a Null rotation with ##l## fixed is $$l^a−>l^a\\ n^a−>n^a+\bar{c}m^a+c\bar{m}^a+c\bar{c}l^a\\ m^a−>m^a+cl^a\\ \bar{m}^a−>\bar{m}^a+\bar{c}l^a$$ Using this transformation, how to prove? $$π−>π+2\bar{c}ϵ+\bar{c}^2κ+D\bar{c}$$ Ref: 2-Spinors by P.O'Donell, p.no, 65
  43. K

    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. D

    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. L

    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. N

    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. M

    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. ChrisVer

    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. B

    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. mnb96

    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...