What is Tensor: Definition and 1000 Discussions

In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

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  1. J

    Differentiating Lagrangian in Tensor Notation

    Homework Statement Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it. Homework Equations Lagrangian density: \mathcal{L} = -\frac{1}{2}...
  2. W

    Orthogonality from infinitesimal small rotation

    Hello buddies, Could someone please help me to understand where the second and the third equalities came from? Thanks,
  3. D

    Understanding the Tensor Product of Two One-Forms in Differential Geometry

    I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...
  4. U

    Energy-Momentum Tensor of Perfect Fluid

    Homework Statement I am given this metric: ##ds^2 = - c^2dt^2 + a(t)^2 \left( dx^2 + dy^2 + dz^2 \right)##. The non-vanishing christoffel symbols are ##\Gamma^t_{xx} = \Gamma^t_{yy} = \Gamma^t_{zz} = \frac{a a'}{c^2}## and ##\Gamma^x_{xt} = \Gamma^x_{tx} = \Gamma^y_{yt} = \Gamma^y_{ty} =...
  5. E

    Cosmological constant term and metric tensor

    Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##. Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?
  6. ShayanJ

    Problem with definition of tensor

    In textbooks, a tensor is usually defined in terms of its transformation properties. But this definition actually seems vague when it comes to checking a set of quantities to see whether they form a tensor or not. Imagine I have four functions and want to see whether they form a 2d 2nd rank...
  7. E

    Baez's vizualisation of Ricci tensor

    I am reading Baez's article http://arxiv.org/pdf/gr-qc/0103044v5.pdf and I do not understand paragraph before equation (10), page 18. Equation (9) will be true if anyone component holds in all local inertial coordinate systems. This is a bit like the observation that all of Maxwell’s equations...
  8. C

    Tensor Variation with Respect to Metric in First Order Formalism

    Homework Statement I'm just wondering if I'm doing this calculation correct? eta and f are both tensors Homework EquationsThe Attempt at a Solution \frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3}...
  9. K

    Multi-scale entanglement renormalization ansatz Tensor network

    as a new proposal for QGhttp://arxiv.org/abs/1502.05385 Tensor network renormalization yields the multi-scale entanglement renormalization ansatz Glen Evenbly, Guifre Vidal (Submitted on 18 Feb 2015) We show how to build a multi-scale entanglement renormalization ansatz (MERA) representation of...
  10. VintageGuy

    Tensor indices (proving Lorentz covariance)

    Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...
  11. fluidistic

    Maxwell stress tensor to calculate force (EM)

    Homework Statement A sphere with dielectric constant ##\varepsilon## and radius R is placed inside a homogenous external electric field ##\vec E_0##. The sphere is divided in 2 hemispheres such that their common interface is orthogonal to the external field. Using the energy-momentum tensor...
  12. P

    Counting Degrees of Freedom in Tensor

    I've been thinking about the number of degrees of freedom in a tensor with n indices in 2-dimensions which is traceless and symmetric. Initially, there are 2^{n} degrees of freedom. The hypothesis of symmetry provides n!-1 number of conditions of the form: T_{i_{1}, \ldots i_{n}}-...
  13. Coffee_

    Question about tensor notation convention as used in SR/GR

    When writing ##A_{a}\text{ }^{b}## one means ''The element on the a-th row and b-th column of the TRANSPOSE of A" right? Such that ##A_{a}\text{ } ^{b}= A^{b}\text{ } _{a}## ? I would just like a confirmation so I'm not learning the convention in a wrong manner.
  14. R

    Is Aijkl a Symmetric Rank 4 Tensor? Proof Needed!

    Homework Statement Let Aijkl be a rank 4 square tensor with the following symmetries: A_{ijkl} = -A_{jikl}, \qquad A_{ijkl} = - A_{ijlk}, \qquad A_{ijkl} + A_{iklj} + A_{iljk} = 0, Prove that A_{ijkl} = A_{klij} Homework EquationsThe Attempt at a Solution From the first two properties...
  15. A

    Ricci tensor equals zero implies flat splace?

    Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help
  16. R

    Calculating Force using the Maxwell Stress Tensor

    Homework Statement Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. Use the Maxwell Stress TensorHomework Equations F=\oint \limits_S \...
  17. B

    Relationship between inverse matrix and inertia tensor?

    Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?
  18. binbagsss

    Valence Tensor of "DVu/Du" Acting on Vector: Analyzing 1 to 1

    Acting upon a vector say, so it is defined as: ##\frac{d}{d\lambda}V^{u}+\Gamma^{u}_{op}\frac{dx^{o}}{d\lambda}V^{p}=\frac{DV^{u}}{D\lambda}## And this can also be written in terms of the covariant derivative, ##\bigtriangledown_{k}## by ##\frac{DV^{u}}{D\lambda}=\frac{d x^{k}}{d \lambda}...
  19. ognik

    Angular momentum of rigid body elements tensor

    Homework Statement I was working through my text on deriving the tensor for Angular momentum of the sums of elements of a rigid body, I follow it all except for one step. Here is a great page which shows the derivation nicely - http://www.kwon3d.com/theory/moi/iten.html I follow clearly to the...
  20. K

    Metric tensor with diagonal components equal to zero

    Hello, Let ##g_{jk}## be a metric tensor; is it possible for some ##i## that ##g_{ii}=0##, i.e. one or more diagonal elements are equal to zero? What would be the geometrical/ topological meaning of this?
  21. binbagsss

    Tensor Density Transformation Law: Order of Jacobian Matrix?

    I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression. As in general Lecture Notes on General Relativity by Sean M. Carroll, 1997 he has the law as ## \xi_{\mu'_{1}\mu'_{2}...\mu'_{n}}=|\frac{\partial...
  22. binbagsss

    Vanishing of Einstein tensor from Bianchi identity

    I'm looking at the informal arguements in deriving the EFE equation. The step that by the bianchi identity the divergence of the einstein tensor is automatically zero. So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##...
  23. C

    Why $H$ is a (1,2) tensor field?

    I have a conceptual question associated with one of the worked examples in my notes. The question is: 'Let ##\nabla## and ##\nabla^*## be connections on a manifold ##M##. Show that ##H(X,Y) = \nabla_X Y - \nabla_X^* Y## where ##X,Y## are vector fields defines a (1,2) tensor on M. To show it...
  24. Ravi Mohan

    Tensor density from wedge product

    Hi, I am studying Sean Carroll's "Lecture notes on General Relativity". In the second chapter he identifies the volume element d^nx on an n-dimensional manifold with dx^0\wedge\ldots\wedge dx^{n-1}. He then claims that this wedge product should be interpreted as a coordinate dependent object...
  25. Galbi

    How can I calculate the inverse of a 4th order tensor?

    Homework Statement I'm looking for how to calculate inverse of the 4th order tensor. That is, A:A-1=A-1:A=I(4) If I know a fourth order tensor A, then how can I calculate A-1 ? Let's just say it is inversible. Homework EquationsThe Attempt at a Solution
  26. nuclearhead

    Best software for tensor manipulation?

    Do you know any good software for manipulating tensors: varying Lagrangians, checking gauge and supersymmetry transformations, etc. ? One that could deal with anti-commuting variables would be good too. One that also supplied group constants for SU(n) etc. would also be useful. I was also...
  27. U

    Raising and lowering Ricci Tensor

    Taken from Hobson's book: How is this done? Starting from: R_{abcd} = -R_{bacd} Apply ##g^{aa}## followed by ##g^{ab}## g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd} g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd} R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd} Applying ##g_{aa}## to both sides...
  28. binbagsss

    Einstein Hilbert action, why varies wrt metric tensor?

    The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know...
  29. R

    Deriving the electromagnetic field strength tensor

    Just one last question today if someone can help. I'm trying to derive the electromagnetic field strength tensor and having a little trouble with (i think) the use of identities, please see below: I understand the first part to get -Ei, but it's the second line of the next bit I don't...
  30. R

    Symmetric rank-2 tensor, relabelling of indices? (4-vectors)

    Homework Statement Homework Equations Relabelling of indeces, 4-vector notation The Attempt at a Solution The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and...
  31. G

    Improved energy-momentum tensor changing dilation operator

    I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...
  32. Q

    Rotational EOM's with non diagonal inertia tensor

    I'm having difficulties understanding how I should calculate the angular velocities of a rigid body when the inertia tensor is given in body coordinates and has off diagonal elements. Let's assume I have an inertia tensor ## I = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{yx} &...
  33. Y

    Help with the variation of the Ricci tensor to the metric

    I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...
  34. teroenza

    Four Tensor Derivatives -- EM Field Lagrangian Density

    Homework Statement Given the Lagrangian density \Lambda = -\frac{1}{c}j^lA_l - \frac{1}{16 \pi} F^{lm}F_{lm} and the Euler-Lagrange equation for it \frac{\partial }{\partial x^k}\left ( \frac{\partial \Lambda}{\partial A_{i,k}} \right )- \frac{\partial \Lambda}{\partial A_{i}} =0...
  35. C

    Stress tensor from action in Landau-Ginzburg field theory

    I would appreciate any help with the following question: I know that for relativistic field theories, the stress tensor can be obtained from the classical action by differentiating with respect to the metric, as is explained on the wikipedia page...
  36. T

    Static universe, spacetime and the stress-energy tensor

    Einstein's static universe obeys ##\rho = 2\lambda##. So, attractive and repelling gravity cancel each other. I'm curious about the spacetime in this universe. Because the scale factor is constant, it seems that neighboring co-moving test particles don't show relative acceleration, thus no...
  37. Z

    How can I solve for a vector in a tensor equation involving dot products?

    Homework Statement I'm currently trying to work through some issues I'm having with tensor and vector analysis. I have an equation of the form $$\textbf{a} \bullet \textbf{b} = \textbf{c} \bullet \textbf{d}$$ where all quantities here are vectors. I want to solve for ##\textbf{b}## by finding...
  38. D

    Eigenvectors of Inertia tensor

    Hi, I've written a little fortran code that computes the three Eigenvectors \vec{v}_1, \vec{v}_2, \vec{v}_3 of the inertia tensor of a N-Particle system. Now I observed something that I cannot explain analytically: Assume the position vector \vec{r}_i of each particle to be given with respect...
  39. HeavyMetal

    Panda"Understand Relationship between Stress-Energy Tensor and Interval

    Hello all, I have a homework question that I am almost 100% sure that I solved, so I do not believe that this post should go into the "Homework Questions" section. This thread does not have to do with the answer to that homework question anyways, but rather a curiosity about whether or not this...
  40. Mentz114

    Tensor Calculus Problem: Simplifying Terms with Index Exchange

    If you don't like indexes, look away now. I got these terms from a tensor calculus program as part of a the transformed F-P Lagrangian. \begin{align} {g}^{b a}\,{g}^{d e}\,{g}^{f c}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c f}\,{g}^{e d}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c...
  41. P

    Finding Tensor in QED: e^+e^- → γγ

    In the process $$e^+e^- \rightarrow \gamma \gamma$$ for which the amplitude can be written as: $$M= \epsilon^*_{1\nu}\epsilon^*_{2\mu}(A^{\mu\nu}+\tilde{A}^{\mu\nu})$$, where $$\epsilon_i$$ is the polarization vector of a photon. How can one find the tensors $$A^{\mu\nu}$$ and...
  42. K

    Question about Metric Tensor: Can g_{rr} be Functions of Coordinate Variables?

    Hello Say, the metric tensor is diagonal, ##g=\mbox{diag}(g_{11}, g_{22},...,g_{NN})##. The (null) geodesic equations are ##\frac{d}{ds}(2g_{ri} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{r}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0## These are ##N## equations containing ##N## partial...
  43. genxium

    What is the parity inversion of antisymmetric tensor

    First by antisymmetric tensor I mean the "totally antisymmetric tensor" like this: ##\epsilon^{\alpha\beta\gamma\delta} = \left\{ \begin{array}{clcl} +1 \;\; \text{when superscripts form an even permutation of 1,2,3,4} \\ -1 \;\; \text{when superscripts form an odd permutation of 1,2,3,4} \\ 0...
  44. U

    Tensor Contraction: Contracting ##\mu## with ##\alpha##?

    What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.
  45. W

    A suggested operational definition of tensors

    The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge they don't make it apparent. So, I'm looking for an operational definition, and suggesting the...
  46. binbagsss

    Energy-momentum tensor radiation-dominated universe.

    I'm looking at 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' Link here:http://arxiv.org/pdf/gr-qc/9712019.pdf Page 221 (on the actual lecture notes not the pdf), where it generalizes that the energy-momentum tensor for radiation - massive particles with velocities tending to...
  47. caffeinemachine

    MHB Natural Isomorphism b/w Dual Spaces Tensor Prod & Multilinear Form Space

    I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...
  48. caffeinemachine

    MHB A Basic Question Regarding the Universal Property of the Tensor Product.

    (All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...
  49. C

    How Does Covariant Differentiation Affect Tensor Fields?

    Homework Statement Let ##T## be a ##(1, 1)## tensor field, ##\lambda## a covector field and ##X, Y## vector fields. We may define ##\nabla_X T## by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = (\nabla_XT)(\lambda, Y ) + T(\nabla_X \lambda, Y ) + T(\lambda, \nabla_X Y ) . $$...
  50. JonnyMaddox

    Program that writes tensor equations out

    Hi, I'm looking for a program that spits out fully summed index equations. For example T_{ii} in, out comes T_{11}+T_{22}+... and so on, with Einstein summation convention.
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