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. C

    I Relation between tensor decomposition and helicity amplitude

    It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices. How do I relate...
  2. olgerm

    I Invariant properties of metric tensor

    Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
  3. D

    I Maxwell Tensor Identity Explained: Deriving Formula 8.23 in Schawrtz's Book

    Hello, In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows: $$-\frac{1}{4}F_{\mu \nu}^{2}=\frac{1}{2}A_{\mu}\square A_{\mu}-\frac{1}{2}A_{\mu}\partial_{\mu}\partial_{\nu}A_{\nu}$$ where: $$F_{\mu\nu}=\partial_{\mu}...
  4. K

    Tensor Force Operator Between Nucleons: Spin & Position

    Homework Statement The tensor force operator between 2 nucleons is defined as ##S_{12}=3\sigma_1\cdot r\sigma_2\cdot r - \sigma_1\cdot \sigma_2##. Where r is the distance between the nucleons and ##\sigma_1##and ##\sigma_2## are the Pauli matrices acting on each of the 2 nucleons. Rewrite...
  5. Math Amateur

    I Differential Forms & Tensor Fiekds .... Browder, Section 13.1

    Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows: In the above text from Browder we read the following: " ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a map...
  6. S

    Commutability of Tensor components , an ambigious situation

    Homework Statement I have been working on defining the transpose in a tensorial view using the kroneck delta tensor.Homework Equations I will use tensor notation The Attempt at a Solution Let Tjk be a 2nd level tensor: (Tjk)TP = Tkj My first attempt is: δjkTjk δkj = Tkj However if tensor...
  7. A

    I Conservation of energy-momentum (tensor)

    For a curve parametrised by ##\lambda## where ##\lambda## is along length of the curve and is 0 at one end point. At each ##\lambda## say tangent vector V and A be the two possible vectors of the tangent space. where ##V=V^\mu e_\mu## and ##A=A^\nu e_\nu##, {e} are the basis vectors. Now ##...
  8. M

    A Riemann Tensor Notation Explained | Choquet-Bruhat GR

    Hello I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors) I don't understand this notation, regarding the use of the semi colon, is it...
  9. U

    Theories without a stress tensor

    Can someone tell me a theory in which the lowest twist operators are not the stress tensor and its derivatives? My aim is to work out the lightcone OPE for the theory and derive bounds like the averaged null energy condition. (as worked out in https://arxiv.org/pdf/1610.05308.pdf)
  10. M

    Nature of displacement and the deformation tensor

    If we have two points P and Q in undeformed material and after deformation they become P' and Q'. The deformation tensor is the derivative of the displacement. What is the displacement? vector PP'? or the change from PQ to P'Q'? is the second question is the strain "change in length". Why the...
  11. M

    Prove that these terms are Lorentz invariant

    Homework Statement Prove that $$\begin{align*}\mathfrak{T}_L(x) &= \frac{1}{2}\psi_L^\dagger (x)\sigma^\mu i\partial_\mu\psi_L(x) - \frac{1}{2}i\partial_\mu \psi_L^\dagger (x) \sigma^\mu\psi_L(x) \\ \mathfrak{T}_R(x) &= \frac{1}{2}\psi_R^\dagger (x)\bar{\sigma}^\mu i\partial_\mu\psi_R(x) -...
  12. Cathr

    I How to derive a symmetric tensor?

    Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX). How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the...
  13. D

    I Energy Tensor Gradients: ∂βTμυ

    I understand, kind of, that ∇μTμυ=0 by conservation or coninuity. What would be ∂βTμυ when β=1,2,3 no time derivative.
  14. M

    Maxwell Stress Tensor: Engineering Question Answered

    Hello! I was talking with a friend today about electrical motors and we started talking about theoretical designs. One question came up which was could the Maxwell Stress Tensor be used to calculate the torque on a rotor of a motor where the airgap is held constant and the magnetic circuit...
  15. fresh_42

    B Tensor Conventions: V^*⊗V^*⊗V (1,2) vs (2,1)

    How do physicists call a tensor of ## V^* \otimes V^* \otimes V##, (1,2) or (2,1)? And which part do they call contravariant and which covariant? I'm just not sure, whether the mathematical definition of funktors apply to the usances in physics. (LUP - tensor)
  16. D

    I Component derivative of a tensor

    This is a simple and maybe stupid question. Can you take a derivative of a vector component with respect to a vector? Or even more generally,can you take the derivative of a component of a tensor with respect to the whole tensor? For instance in the cauchy tensor could you take the xx component...
  17. A

    I Contravariant derivative of tensor of rank 1

    If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk
  18. A

    How Can We Derive the General Stress Tensor Without Assuming Linear Media?

    Homework Statement Hi everyone! My name is Alexandra, and I'm new in this forum. I am trying to determine the mentionated tensor without the assumption of linear media or vacuum ( ## \textbf{D} = \epsilon \textbf{E} ## and ## \textbf{B} = \mu \textbf{H} ##). What I want to obtain is the...
  19. V

    I Riemann Tensor: Questions & Geometric Interpretation

    Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the...
  20. H

    I Understanding the stress-energy tensor

    I have trouble understanding some terms in the stress-energy-tensor. For instance T^(12) stands for the flux of the x-component of momentum in the y-direction. But what does it means for the x-component of momentum to flow in the y direction? Since momentum is a vector should't the x-component...
  21. N

    A Symmetry of the permittivity tensor of lossless media

    I read in various sources (such as page 8 of these notes) that the dielectric permittivity tensor of a lossless medium is always symmetric. I am wondering how this can be the case, when: Phase accumulation in the medium could in theory depend on direction Coordinate system may be rotated to...
  22. K

    I Why is the Energy Momentum Tensor of a Perfect Fluid a Tenso

    The energy momentum tensor of a perfect relativistic fluid is given by $$T^{\mu\nu} = (\rho + p)u^\mu u^\nu + p g^{\mu\nu}$$ I don't understand why this is a tensor, i.e. why it transforms properly under coordinate changes. ##u^\mu u^\nu## and ##g^{\mu\nu}## are tensors, so for ##T^{\mu\nu}##...
  23. T

    Is this derivative in terms of tensors correct?

    Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...
  24. saadhusayn

    Finding the Ricci tensor for the Schwarzschild metric

    I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor. The given distance element is $$ ds^2 = e^{2 \lambda} dt^2 -...
  25. M

    Quantum Teleportation Homework: Deriving EPR Pair & Measuring Spin 1/2 Particles

    Homework Statement This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation. In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as $$ \vert...
  26. Andrea B DG

    A Electromagnetic tensor and time reversal

    Consider equation (2.7.8) page 42 in the book Gravitation and Cosmology by Weinberg F' αβ = Λαγ Λβδ Fγδ Now consider the time reversal Lorenz transformation Λμν = 0 if μ ≠ ν, 1 if μ = ν = 1..3 and -1 if μ = ν = 0 then F' 00 = 0 F' 0i = -F 0i F' ij = F ij Using equation (2.7.5) of the same book...
  27. sams

    Questions Regarding the Inertia Tensor

    In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor: 1.The authors made the following statement: "neither V nor ω...
  28. K

    A Definition of Tensor and.... Cotensor?

    Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...
  29. Chromatic_Universe

    Specific proof of the Riemann tensor for FRW metric

    Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...
  30. Digital_lassitude

    Is it possible to express friction force as a tensor?

    Homework Statement Consider the equation for the friction force Ff = m FN. is it possible to express the friction force as a tensor? If so, what rank tensor is it, and what are the ranks of the tensor m and the normal force FN? Homework Equations Ff = mFNThe Attempt at a Solution [/B] So I...
  31. M

    I Convert Metric Tensor to Gravity in GR

    I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa. Thanks!
  32. K

    MHB Understanding Tensor Calculations: Exploring Equations 4.74 and 4.76

    Two questions have me a bit stumped. I am given: eqn 4.74 $J^{i}_{i'} J^{i'}_{j} =\delta^{i}_{j}$ and eqn 4.76 $J^{i'}_{i} J^{i}_{j'} =\delta^{i'}_{j'}$ Problem 47: Derive equation 4.76 from 4.74 by multiplying both sides by $J^j_{j'}$. I have gone in a bunch of circles but can't get...
  33. S

    A Stressing Over Stress Tensor Symmetry in Navier-Stokes

    How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations: Behavior of a Vorticity Influenced Asymmetric Stress Tensor In Fluid Flow http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf...
  34. C

    Is there any tensor that gives the radius of a sphere?

    Can I calculate a tensor of a system( lots of particles) shaped like a sphere, then get exactly the radius of the system? (I want to get lengths of three axes of ellipsoid, and I'm trying to examine the way with a sphere. )
  35. F

    I Get Relation from Stress-Energy Tensor Def.

    Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$ with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...
  36. B

    Is the Energy Momentum Tensor for Scalar Fields Always Symmetric?

    Homework Statement Show that if the Lagrangian only depends on scalar fields ##\phi##, the energy momentum tensor is always symmetric: ##T_{\mu\nu}=T_{\nu\mu}## Homework Equations ##T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L## The Attempt at a...
  37. B

    Stress energy tensor transformation

    Homework Statement Show that if you add a total derivative to the Lagrangian density ##L \to L + \partial_\mu X^\mu##, the energy momentum tensor changes as ##T^{\mu\nu} \to T^{\mu\nu}+\partial_\alpha B^{\alpha\mu\nu}## with ##B^{\alpha\mu\nu}=-B^{\mu\alpha\nu}##. Homework EquationsThe Attempt...
  38. M

    Solving the Einstein Gravity Tensor for the Newton Potential

    Homework Statement The Lagrangian density for the ##h=h^{00}## term of the Einstein gravity tensor can be simplified to: $$L=-\frac{1}{2}h\Box h + (M_p)^ah^2\Box h - (M_p)^b h T$$ The equations of motion following from this Lagrangian looks roughly like (I didn't calculate this, they are given...
  39. N

    Inertia tensor of cone around its apex

    Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB. % Symbolic variables syms r z theta R h M; % R (Radius) h(height) M(Mass) % Ixx unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...
  40. Another

    Help with Maxwell stress tensor

    << Mentor Note -- OP has been reminded to use the Homework Help Template when posting schoolwork questions >> my think if ## \hat{r} = \sin(θ) \cos( φ) \hat{x} +\sin(θ) \sin( φ) \hat{y} +\cos(θ) \hat{z} ## ## da = R^2 \sin(θ) dθdφ \hat{r} = da_{x} \hat{x} + da_{x} \hat{y} + da_{z} \hat{z}##...
  41. kasoll

    Inertia tensor v.s pincipal axes moment of inertia

    Is there a method to calculate inertia tensor form principal axes moment of inertia? Like now we have moment of inertia: (Ix,Iy,Iz)=(20,18,25), and hot to calculate the inertia tensor like (Ixx,Ixy,Ixz Iyx,Iyy,Iyz, Izx,Izy,Izz)? I have read about this page several times, but still have no idea.
  42. E

    A Vec norm in polar coordinates differs from norm in Cartesian coordinates

    I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
  43. K

    MHB Skew-Symmetric Tensor: Computing [W]e and Axial Vector w

    If W is a skew-symmetric tensor, (i) Write down the most general form of [W]e. (ii) Show that there exist a vector w in R3 such that Wx = w*x for each x in R3. Such a vector w is called the axial vector of W. (iii) Use part (ii) to deduce that x.Wx = 0 for any x in R3. (iv) If W has axial vector...
  44. snoopies622

    I How to keep the components of a metric tensor constant?

    I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...
  45. Prez Cannady

    I Confused by this result for the tensor product of two vectors

    Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...
  46. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

    I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
  47. C

    I Lorentz Group: Tensor Representation Explained

    I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...
  48. Orodruin

    Insights The 10 Commandments of Index Expressions and Tensor Calculus - Comments

    Greg Bernhardt submitted a new PF Insights post The 10 Commandments of Index Expressions and Tensor Calculus Continue reading the Original PF Insights Post.
  49. omega_minus

    I Deriving Maxwell's Equations from Field Tensor (Griffith 4ed)

    Hello, I am reading Griffith's "Introduction to Electrodynamics" 4ed. I'm in the chapter on relativistic electrodynamics where he develops the electromagnetic field tensor (contravariant matrix form) and then shows how to extract Maxwell's equations by permuting the index μ. I am able to...
  50. shahbaznihal

    I Solving Tensor Calculus Question from Schutz Intro to GR

    I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
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