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. M

    Why does the tensor product in QM produce unentangled states?

    In QM the tensor product of two independent electron's spin state vectors represents the product state which represents the possible unentangled states of the pair. I don't understand why the tensor product produces that result. |A⟩=|a⟩⊗|b⟩
  2. M

    Learn Wigner Rotation, Tensor Operator & Two-Particle Helicity State

    Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.
  3. L

    Tensor techniques in $3\otimes\bar 3$ representation of su(3)

    Hi everyone! I would like to ask you a very basic question on the decomposition 3\otimes\bar 3=1\oplus 8 of su(3) representation. Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form O^{y}_{x} where upper index belongs to the $\bar 3$ rep and the...
  4. B

    Geometrical meaning of Weyl tensor

    Can anyone give me a geometrical interpretation of the weyl curvature tensor?
  5. andrewkirk

    Can contraction of a tensor be defined without using coordinates?

    All but one of the tensor operations can be defined without reference to either coordinates or a basis. This can be done for instance by defining a ##(^m_n)## tensor over vector space ##V## as a multi-linear function from ##V^m(V^*)^n## to the background field ##F##. This allows us to define...
  6. WannabeNewton

    Killing fields as eigenvectors of Ricci tensor

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  7. T

    Proving that the det of the Lorentz tensor is +1 or -1 (MTW p 89)

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  8. D

    Definition of the Einstein Tensor

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  9. fluidistic

    I don't understand the notation (tensor?, not even sure)

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  10. I

    Average of a tensor composed of unit vectors

    Hello, I'm studying The Theory of Polymer Dynamics by Doi and Edwards and on page 98 there is a tensor, defined as a composition of two identical unit vectors pointing from the monomer n to the monomer m: \hat{\textbf{r}}_{nm}\hat{\textbf{r}}_{nm} As far as I understood, the unit vectors...
  11. P

    Surface Integration of vector tensor product

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  12. P

    Why Do These Riemann Tensor Terms Cancel Each Other Out?

    I was working on the derivation of the riemann tensor and got this (1) ##\Gamma^{\lambda}_{\ \alpha\mu} \partial_\beta A_\lambda## and this (2) ##\Gamma^{\lambda}_{\ \beta\mu} \partial_\alpha A_\lambda## How do I see that they cancel (1 - 2)? ##\Gamma^{\lambda}_{\ \alpha\mu}...
  13. P

    Energy Momentum Tensor - EM - Derivative

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  14. S

    Curvature Tensor: Non-Zero in Local Inertial Frame

    hi In a local inertial frame with g_{ij}=\eta_{ij} and \Gamma^i_{jk}=0. why in such a frame, curvature tensor isn't zero? curvature tensor is made of metric,first and second derivative of metric.
  15. S

    Proving When a Tensor is 0: M \cong M \otimes K

    Let M be a module over the commutative ring K with unit 1. I want to prove that M \cong M \otimes K. Define \phi:M \rightarrow M \otimes K by \phi(m)=m \otimes 1. This is a morphism because the tensor product is K-linear in the first slot. It is also easy to show that the map is surjective...
  16. T

    Metric tensor at the earth surface

    I want to find the ricci tensor and ricci scalar for the space-time curvature at the Earth surface. Ignoring the moon and the sun. I have used the scwharzschilds metric, but then the ricci tensor and the scalar where equal to zero.
  17. N

    Poisson bracket and Electric and Magnetic Weyl tensor in GR

    In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory. I have tried to look at the Poisson bracket, but the...
  18. E

    What is the Riemann Curvature Tensor for Flat and Minkowski Space?

    Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...
  19. I

    Independency of the frame of reference of the strain rate tensor

    I've got a problem regarding tensors. Premise: we are considering a fluid particle with a velocity \mathbf{u} and a position vector \mathbf{x}; S_{ij} is the strain rate tensor, defined in this way: \displaystyle{S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial...
  20. T

    Proof: Pauli-Lujanski tensor

    I want to proof that [M\mu\nu,W\sigma]=i(g\nu\sigmaW\mu-g\mu\sigmaW\nu) I can reduce this expression but I can't find the correctly answer. Thanks!
  21. F

    Rank-2 tensor: multiple definitions

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  22. 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...
  23. Z

    Weyl tensor in 2 dimensions- confused

    hello, The Weyl tensor is: http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif In 2 dimensions , the Riemann tensor is (see MTW ex 14.2): Rabcd = K( gacgbd - gadgbc ) [R] Now the Weyl tensor must vanish in 2 dimensions. However, working with the g g = [-1 0 0...
  24. J

    Static, spherically symmetric Maxwell tensor

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  25. B

    Operation to make an (m+n)th rank tensor of rank-m and rank-n tensors

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  26. Philosophaie

    Finding the 4x4 Cofactor of a Covariant Metric Tensor g_{ik}

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  27. P

    Difference between tensor product and direct product?

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  28. S

    What is the Form of Riemann Tensor in 3D?

    hi Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it. see for example Geometry,Topology and physics By Nakahara Ch.7 In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel...
  29. C

    Spherical tensor operators' commutation with lowering/raising operator

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  30. nomadreid

    Which of these statements about tensor products is incorrect?

    I have read the following three simplifications in various places, but together they give a contradiction, so at least one of them must be an oversimplification. Which one? (a) Interaction between two systems A and B is described by A\otimesB (b) An entangled state C is a pure state, and...
  31. P

    Tensor summation and components.

    Hello, I would very much like someone to please clarify the following points concerning tensor summation to me. Suppose the components of a tensor Ai j are A1 2 = A2 1 = A (or, in general, Axy = Ayx = A), whereas all the other components are 0. Is this a symmetrical tensor then? How may Ai j be...
  32. P

    Transforming Tensor Components with Coordinate Systems

    Hi, Homework Statement The components of the tensor Ai j are A1 2 = A2 1 = A, whereas all the other components are zero. I am asked to write A(BAR)i j, following a transformation to a new coordinate system, given that ∂q(BAR)k/∂qn = Rnk. I am expected to write my answer in terms of R...
  33. Markus Hanke

    Tensor Index Notation Question

    I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other A_{\mu }^{\nu } and being "spaced" as in A{^{\nu }}_{\mu } I seem to remember that I once read that there is indeed a difference, but I can't remember what it...
  34. R

    Solve the Stress Tensor Problem Now

    http://im32.gulfup.com/u6n3f.jpg
  35. Fredrik

    What is the issue with tensor terminology and variable slot order?

    Definitions like this one are common in books: For all ##k,l\in\mathbb N##, a multilinear map $$T:\underbrace{V^*\times\cdots\times V^*}_{k\text{ factors}}\times \underbrace{V\times\cdots\times V}_{l\text{ factors}}\to\mathbb R$$ is said to be a tensor of type ##(k,l)## on ##V##. Lee calls this...
  36. S

    Stress tensor rotation/shear stress

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  37. Mandelbroth

    Metric Tensor on a Mobius Strip?

    I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :tongue: Atrocious comedy aside, Spivak provides a...
  38. T

    What is the relationship between scalar fields and tensor fields?

    I've PMd some of you with this question, but I got some conflicting replies or no replies at all lol, so I'm posting it here. I also did a Google search and found this which I'm almost sure answers my question, but I just want to confirm with you guys: ''In general, scalar fields are referred...
  39. L

    Confusion with Einstein tensor notation

    Homework Statement I'm confused about writing down the equation: \Lambda \eta \Lambda^{-1} = \eta in the Einstein convention. Homework Equations The answer is: \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma} However it's strange because there seems...
  40. P

    Ricci tensor of the orthogonal space

    While reading this article I got stuck with Eq.(54). I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. Is the...
  41. H

    Tensor software for General Relativity

    I have used GRTensorII and Cadabra for some time. And I think Cadabra have great potential for GR. But the current vision of Cadabra only deals with abstract tensor analysis, not with writing out of explicit components. So ,(eq :)when I try to check my final tensor expressions of solutions of...
  42. Y

    Reducibility tensor product representation

    Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...
  43. Y

    Stuck on proof irreducible spherical tensor operator

    Hi everyone, I'm stuck on proving that a certain operator is an irreducible spherical tensor operator. These are tensor operators T^{k}_{q} with -k \leq q\leq k satisfying \mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'} where...
  44. E

    Wrong signs Ricci tensor components RW metric tric

    Hi, I am working through GR by myself and decided to derive the Friedmann equations from the RW metric w. ( +,-,-,-) signature. I succeeded except that I get right value but the opposite sign for each of the Ricci tensor components and the Ricci scalar e.g. For R00 I get +3R../R not -3R../R . I...
  45. Q

    Lorentz transformation matrix applied to EM field tensor

    In a recent course on special relativity the lecturer derives the Lorentz transformation matrix for the four vector of position and time. Then, apparently without proof, the same matrix is used to transform the EM field tensor to the tensor for the new inertial frame. I am unclear whether it...
  46. R

    Tensor product: commutator for spin

    Homework Statement [S2 total, Sz ∅1] Homework Equations S2 total = S2∅1+ 1∅S2+2(Sx∅Sx+Sy∅Sy+ Sz∅Sz) The Attempt at a Solution I calculated it in steps: (1∅Sx 2 +Sx 2∅1) * Sz ∅1 =[S2x, Sz] ∅1 + Sz∅Sx 2 =-h_cut i (SxSy+SySx)∅1 + Sz∅Sx 2 Is it correct way of doing it? I...
  47. M

    Elastic Energy Momentum Tensor and Defects

    Hi All, I am reading the seminal paper by Eshelby on the elastic energy-momentum tensor, which I attach for your convenience. It is all beautiful but equation 4.4 at the beginning. He considers a surface S in the undeformed configuration of a body. The surface is translated by a vector u to a...
  48. Y

    Tensor product of Hilbert spaces

    Hi everyone, I don't quite understand how tensor products of Hilbert spaces are formed. What I get so far is that from two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 a tensor product H_1 \otimes H_2 is formed by considering the Hilbert spaces as just vector spaces H_1 and H_2...
  49. 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...
  50. 4

    Question about value of the metric tensor and field strength

    Is it the value of the metric tensor that determines the strength of a gravitational field at a specific point in spacetime?
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