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. Gene Naden

    Geometric Version of Maxwell Equation related to Tensor Dual

    Homework Statement From Misner, Thorne and Wheeler's text Gravitation (MTW), exercise 3.15: Show that, if F is the EM field tensor, then ##\nabla \cdot *F## is a geometric, frame-independent version of the Maxwell equation...
  2. T

    I How does the stress-energy tensor act on gravity?

    How do the components of the stress-energy tensor act on gravity regarding a) the FRW-universe? b) a solid ball? In a FRW-universe ##\rho + 3P## determines the second derivative of the scale factor. So, there are no non-diagonal components. Just theoretically, if the perfect fluid was...
  3. S

    I Representing conversion of (1,1) tensor to (2,0) tensor

    A non-degenerate Hermitian form ##(.|.)## on a vector space ##V## can be identified with a map ##L:V \to V^*## such that ##L(v)=\tilde{v}## and ##\tilde{v}(w) \equiv (v~|~w)##. Suppose we want to convert a vector ##v## to a dual vector ##\tilde{v}##. In terms of matrices, we can just construct...
  4. W

    Proving the symmetry property of Riemann curvature tensor

    Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...
  5. saadhusayn

    Divergence of the energy momentum tensor

    I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e. $$ \partial_{\mu} T^{\mu \nu} = 0$$ where $$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...
  6. granzer

    I What is the magnitude of a tensor?

    I know that a vector is a tool to help with quantities that have both a magnitude a direction. At a given point in space, a vector has a particular magnitude and direction and if we take any other direction at the same point we can get a projection of this vector in that direction. Tensor is a...
  7. CantorsLuck

    A Momentum Energy tensor and Wilson Loop in Yang-Mills Theory

    Hello Everyone. I Was Wondering how excatly the Gauge invariance of the trace of the Energy-momentum tensor in Yang-Mills theory connects with the trace of an Holonomy. To be precise in what I'm asking: The Yang-Mills Tensor is defined as: $$F_{\mu \nu} (x) = \partial_{\mu} B_{\nu}(x)-...
  8. BookWei

    Field strength tensor for a moving charged particle

    Hi, I am studying Chapter14 in Jackson. My attached file is about field strength tensor. My question is how can I obtain the radiation and the non-radiation terms in the field strength tensor for a moving charged particle. Many thanks.
  9. LesterTU

    How to evaluate the effective mass tensor (band structure)

    Homework Statement The energy-band dispersion for a 3D crystal is given by $$E(\mathbf k) = E_0 - Acos(k_xa) - Bcos(k_yb) - Ccos(k_zc)$$ What is the value of the effective mass tensor at ## \mathbf k = 0 ##? Homework Equations The effective mass tensor is given by $$ \left( \frac{1}{m^*}...
  10. t_r_theta_phi

    I Intuitive explanation for Riemann tensor definition

    Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...
  11. Z

    I Components of Riemann Tensor: 4 Indexes, 16x16 Matrix

    Hello, Riemann tensor ##R^i_{jkl}## 4 indexes, and it should be matrix 16x16 in spacetime if we have time coirdinate - 0 and space coordinates -1,2,3. But how should I write the components to matrix? For example ##\begin{pmatrix}R^0_{000} & R^1_{000} & R^2_{000} ... \\ R^0_{100} & R^1_{100} &...
  12. Z

    I Calculating the Ricci tensor on the surface of a 3D sphere

    Hello, I'm trying to calculate Christoffel symbols on 2D surface of 3D sphere, the metric tensor is gij = diag {1/(1 − k*r2), r2}, where k is the curvature. I derived it using the formula for symbols of second kind, but I think I've made mistake somewhere. Then I would like to know which of the...
  13. J

    Band diagram, conductivity tensor

    Hello! Does anyone have an idea of how can I obtain information from a band diagram about the directions along which the system conducts best and worst ? Thank you in advanced! :)
  14. K

    Energy-momentum tensor from a Lagrangian density?

    Homework Statement I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry. I want to apply this to a specific instance but I am unsure of the approach. Homework Equations for a field...
  15. H

    Which strain tensor is used by Abaqus?

    Hi all, I have some doubts regarding the strain tensors Abaqus uses for the case of Geometrical non linear analysis. In the case of geometrically linear analysis, all the strain tensors will be equal to engineering strain, So it doesn't matter which strain tensor Abaqus uses. But for...
  16. binbagsss

    Tensor Covariant Derivative Expressions Algebra (Fermi- Walk

    Homework Statement Hi I am looking at part a). Homework Equations below The Attempt at a Solution I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this. So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...
  17. C

    I Riemann curvature tensor derivation

    Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...
  18. P

    A Compute Normal Forces on Box Sides via Stress-Energy Tensor

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  19. J

    How Can Crystal Symmetry Affect Conductivity Tensor Components?

    Hello guys! I have to solve a problem about crystal symmetry, but I am very lost, so I wonder if anyone could guide me. The problem is the following: Using semiclassical transport theory the conductivity tensor can be defined as: σ(k)=e^2·t·v_a(k)·v_b(k) Where e is the electron charge, t...
  20. M

    B G11 Metric Tensor: What is it & How Does it Work?

    What is g11? I am very curious, can someone briefly describe what the metric tensor is, please?
  21. kvothe18

    I Mean value of the nuclear tensor operator

    Does anyone know how can you prove that the mean value of the tensor operator S12 in all directions r is zero? S12 : http://prntscr.com/j3gn40 where s1, s2 are the spin operators of two nucleons.
  22. Marcus95

    Time Derivative of Rank 2 Tensor Determinant

    Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...
  23. binbagsss

    Energy-momentum tensor perfect fluid raise index

    Homework Statement This should be pretty simple and I guess I am doing something stupid? ##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}## compute ##T^u_v##: ##T^0_0=\rho, T^i_i=-p##Homework Equations ##U^u=\delta^t_u## ##g_{uv}## is the FRW metric,in particular ##g_{tt}=1## ##g^{bu}T_{bv}=T^u_v## ##...
  24. A

    Electromagnetic Tensor: Calculating $\det{F^{\mu}}_\nu$

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  25. e2m2a

    I Vector and Scalar Tensor Invariance

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  26. e2m2a

    A Dark Matter, Energy-Momentum Tensor & Galaxies

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  27. dRic2

    I Significance of the viscous stress tensor

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  28. 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...
  29. Gene Naden

    I Derivatives of EM Four-Potential: Euler-Lagrange to $\nabla \times B$

    So the Euler-Lagrange equations give ##\partial _\mu ( \partial ^\mu A^\nu - \partial ^\nu A^\mu ) = J^\nu## with ##B=\nabla \times A##. I want to convert this to ##\nabla \times B - \frac{\partial E}{\partial t} = \vec{j}##. I reckon I am supposed to use the Minkowski metric to raise or lower...
  30. L

    A Tensor symmetries and the symmetric groups

    In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...
  31. Vectronix

    Polarization-Magnetization Tensor

    Please forgive me if I chose the wrong thread level. I don't think this is an undergrad topic but I'm not sure. I'm looking for some info about the polarization-magnetization tensor; I can't seem to find it anywhere.
  32. I

    I Vectors in Minkowski Space & Parity: Checking the Effect

    It is known that vectors change them sing under the influence of parity when ##(x,z,y)## change into ##(-x,-z,-y)## $$P: y_{i} \rightarrow -y_{i}$$ where ##i=1,2,3## But what about vectors in Minkowski space? Is it true that $$P: y_{\mu} \rightarrow -y_{\mu}$$ where ##\mu=0,1,2,3##. If yes how...
  33. I

    I Pseudotensors in different dimensions

    In this topic https://physics.stackexchange.com/questions/129417/what-is-pseudo-tensor one answer was the next: The action of parity on a tensor or pseudotensor depends on the number of indices it has (i.e. its tensor rank): - Tensors of odd rank (e.g. vectors) reverse sign under parity. -...
  34. V

    I Proving Effects of Stress-Energy Tensor on Curvature

    Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...
  35. G

    Construction of metric from tensor products of vectors

    1. The metric ##g_{\mu \nu}## of spacetime shall be constructed from tensor products of vectors (relevant are the unit vectors in the respective directions). One such vector shall be called ##A##. Homework Equations ##g_{\mu \nu} = \lambda \frac{A_\mu A_\nu}{g^{\alpha \beta} A_\alpha A_\beta}...
  36. Torg

    A Divergence of (covaraint) energymomentum tensor

    whyT^[ab][;b]≠T_[ab][;b] for spatially flat FLWR cosmology ((ds)^2=(c^2)* (dt)^2-a(t)^2[(dx)^2+(dy)^2+(dz)^2])? τ[ab][/;b] gives the right answer, but not τ[ab][/;b]. (T^(ab) or T_(ab)) contra-variant and co-variant energy momentum tensor of perfect fluid (;) covariant derivative, (c) spped of...
  37. BiGyElLoWhAt

    I How to fill the stress energy tensor for multi body systems

    Say I wanted to set up EFE for the Earth and moon. How do I actually go about filling the stress energy tensor? I'm referencing the wikipedia page. So the time-time should be approximately E/c^2V, so for the Earth moon system ##T_{00} = \frac{3}{4\pi r_E^3}\frac{1}{c^2}(M_Ec^2 + 2/5...
  38. S

    B Metric Tensor and The Minkowski metric

    Hi, I have seen the general form for the metric tensor in general relativity, but I don't understand how that math would create a Minkowski metric with the diagonal matrix {-1 +1 +1 +1}. I assume that using the kronecker delta to create the metric would produce a matrix that has all positive 1s...
  39. vibhuav

    I Requesting clarification about metric tensor

    I am a little bit confused about the metric tensor and would like some feedback before I proceed with my learning of GR. So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come...
  40. P

    A Exploring the Stress-Energy Tensor of a Perfect Fluid

    I believe this thread is sufficiently different from one that was recently closed to not violate any guidelines, though there are unfortunately some similarities as the closed thread sparked the questions in my mind. If we look at the stress energy tensor of a perfect fluid in geometric units...
  41. S

    Stiffness Stress Tensor Question

    Homework Statement I am given c11, c12, and c44. What is poissons ratio ν and the E modulus E [100] for a single crystal for uniaxial strain in [100] (if Fe is isotropic)? ii) What is the anisotropy factor A? (iii) There is: sigma=[100 0 0; 0 100 0; 0 0 0]Mpa What is the transverse strain in...
  42. I

    Operation with tensor quantities in quantum field theory

    I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera. I...
  43. vibhuav

    I Evaluating metric tensor in a primed coordinate system

    I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...
  44. R

    Contraction of a tensor to produce scalar

    Homework Statement Explain how it is possible to perform a contraction of the tensor ##T^{\beta \gamma}_{\delta \epsilon}## in order to produce a scalar T Homework EquationsThe Attempt at a Solution $$T^{\beta \gamma}_{\delta \epsilon}T_{\beta \gamma}^{\delta \epsilon}=T$$ Not sure if that is...
  45. N

    B Tensor Product, Basis Vectors and Tensor Components

    I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
  46. Kushwoho44

    Understanding the Cauchy Stress Tensor: Clearing Up My Confusion

    I have been trying to fully grasp the concept of the Cauchy stress tensor and so I thought I'd make a post where I clear up my confusion. There may be subsequent replies as I pose more questions. I am specifically confused at how the stress tensor relates to the control volume in the image...
  47. C

    A Faraday Tensor Conflicts: MTW vs. Wikipedia

    Hello, I've found that the Faraday Tensor with both indeces down has in the first line, in MTW Gravitation book (pg 74, eq 3.7), minus the electrical field, while in Wikipedia we find that it is plus the electrical field. Which one is right? Does it depend on the signature of the metric?
  48. P

    I Non-zero components of Riemann curvature tensor with Schwarzschild metric

    I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...
  49. Van Ladmon

    A proof about tensor invariants

    Homework Statement How to proof the following property of tensor invariants? Where: ##[\mathbf{a\; b\; c}]=\mathbf{a\cdot (b\times c)} ##, ##\mathbf{T} ##is a second order tensor, ##\mathfrak{J}_{1}^{T}##is its first invariant, ##\mathbf{u, v, w}## are vectors. Homework Equations...
  50. S

    I Understanding Kunneth Formula and Tensor Product in r-Forms

    Hello! Kunneth fromula states that for 3 manifolds such that ##M=M_1 \times M_2## we have ##H^r(M)=\oplus_{p+q=r}[H^p(M_1)\otimes H^q(M_2)]##. Can someone explain to me how does the tensor product acts here? I am a bit confused of the fact that we work with r-forms, which are by construction...
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