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.

View More On Wikipedia.org
Recent contents Top users
  1. K

    I Understanding the Derivation of the Metric Tensor

    Hello, I have a question regarding the first equation above. it says dui=ai*dr=ai*aj*duj but I wonder how. (sorry I omitted vector notation because I don't know how to put them on) if dui=ai*dr=ai*aj*duj is true, then dr=aj*duj |dr|*rhat=|aj|*duj*ajhat where lim |dr|,|duj|->0 which means...
  2. V

    I Metric tensor derived from a geodesic

    Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
  3. L

    Partial traces of density operators in the tensor product

    Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...
  4. B

    I How do i find the eigenvalues of this tough Hamiltonian?

    I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...
  5. S

    Can transformation coefficients be interchanged in symmetric tensors?

    Homework Statement The lecture notes states that if ##T_{ij}=T_{ji}## (symmetric tensor) in frame S, then ##T'_{ij}=T'_{ji}## in frame S'. The proof is shown as $$T'_{ij}=l_{ip}l_{jq}T_{pq}=l_{iq}l_{jp}T_{qp}=l_{jp}l_{iq}T_{pq}=T'_{ji}$$ where relabeling of p<->q was used in the second...
  6. arpon

    I Is Second rank tensor always tensor product of two vectors?

    Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...
  7. S

    A Extracting Tensor Algebra Term with SU(N) Generators and Numbers

    Consider the expression $$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$ where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers. How...
  8. davidge

    I Green's theorem in tensor (GR) notation

    Hi. I was trying to translate the divergence theorem and the Green's theorem to tensor notation that we use in Relativity. For the divergence theorem, it was easy (please tell me if I'm wrong in the below derivation). I'm using the standard electromagnetic tensor ##F_{\mu \nu}## in place of the...
  9. T

    Find the Riemannian curvature tensor component

    Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...
  10. binbagsss

    General Relativity, identity isotropic, Ricci tensor

    Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...
  11. S

    I Number of independent components of the Riemann tensor

    I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...
  12. R

    B Metric tensor of a perfect fluid in its rest frame

    The stress-energy tensor of a perfect fluid in its rest frame is: (1) Tij= diag [ρc2, P, P, P] where ρc2 is the energy density and P the pressure of the fluid. If Tij is as stated in eq.(1), the metric tensor gij of the system composed by an indefinitely extended perfect fluid in...
  13. C

    How to prove that something transforms like a tensor?

    Homework Statement I have several problems that ask me to prove that some quantity "transforms like a tensor" For example: "Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form...
  14. D

    A Find EM Potential from EM Tensor | Math Solutions

    Hello, so my question is, if for some metric, we have found (somehow) Fμν, and we know that: Fμν=∂μAν-∂νAμ, how do we find Aν? I tried solving the differential system after imposing the Lorentz gauge ∂μAμ=0 but still, without some initial guess about which components of A are zero, the system...
  15. S

    Inertia tensor of a body rotating about 3 axes

    Homework Statement Hello, I know about the inertia tensor about one axis, but how about a body that rotates around 3 axis x,y and z such as a spacecraft with changes in the attitude. Thanks for you help. Homework EquationsThe Attempt at a Solution
  16. zwierz

    A Is the Gradient a Covector and the Cross Product a Pseudovector?

    We have got a disagreement with fresh_42 in https://www.physicsforums.com/threads/the-pantheon-of-derivatives-part-ii-comments.908009/#post-5718965 So I would like to ask specialists in differential geometry for a comment 1) gradient of a function defined as follows $$\nabla...
  17. C

    I Tensor integrals in dimensional regularisation

    Consider a d dimensional integral of the form, $$\int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma} \ell^{\mu}}{D}\,\,\,\text{and}\,\,\, \int \frac{d^d \ell}{(2\pi)^d} \frac{\ell^{\sigma}}{D}$$ where ##D## is a product of several propagators. One can reduce this to a sum of scalar integrals by...
  18. Vitani11

    I Difference between tensor and vector?

    δij is the Kronecker delta - is this considered a tensor or vector? I know it means the identity when i=j so I'm going to guess tensor because it's a matrix rather than just a vector but I want to make sure. A matrix is a rank 2 tensor and a vector is a rank 1 and a scalar is a rank 0? How does...
  19. D

    A Difference Between Outer and Tensor

    Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##. Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##. In Dirac's Bra-Ket notation, this is written as...
  20. M

    Which strain tensor is used in Föppl–von Kármán equations?

    Hi, I have a question concerning the von Kármán equations. I want to better understand the compatibility relation. The wikipedia article states that: https://en.wikipedia.org/wiki/F%C3%B6ppl%E2%80%93von_K%C3%A1rm%C3%A1n_equations "The components of the three-dimensional Lagrangian Green strain...
  21. S

    I Deducing the tensorial structure of a tensor

    Consider an expression of the following form: $$I^{\mu\nu}(r) = \int d^{3}k\ \ d^{3}l\ \ \delta^{4}(r-k-l)\ (g^{\mu\nu}k\cdot{l}+k^{\nu}k^{\mu}-k^{\mu}l^{\nu})$$ ##I^{\mu\nu}## must be of the form $$I^{\mu\nu}(r) = Ar^{\mu}r^{\nu} + B\eta^{\mu\nu},$$ where ##A## and ##B## are constants. How...
  22. T

    A Hamiltonian with a tensor product - a few basic questions

    I am given a hamiltonian for a two electron system $$\hat H_2 = \hat H_1 \otimes \mathbb {I} + \mathbb {I} \otimes \hat H_1$$ and I already know ##\hat H_1## which is my single electron Hamiltonian. Now I am applying this to my two electron system. I know very little about the tensor product...
  23. davidge

    I Riemann tensor in 3d Cartesian coordinates

    Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...
  24. mhob

    A Hypermomentum & Belinfante-Rosenfeld: Same Object?

    Are they the same object?
  25. Kara386

    I Exploring Tensor Products: Index Notation

    We've been learning about tensor products. In particular, we've been looking at index notation for the tensor products of matrices like these: ## \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)## And ## \left( \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22}...
  26. arivero

    I Bootstraping a space from its tensor square

    By space, I mean a vector space which could be a representation of a group or even have some expanded algebraic structure. So I am not sure if this question goes here or in the Algebra subforum. Consider the tensor square r\otimes r of an irreducible group representation r with itself, and...
  27. F

    I What is the concept of tensor product and how is it used in mathematics?

    Hello, I have encountered the concept of tensor product between two or more different vector spaces. I would like to get a more intuitive sense of what the final product is. Say we have two vector spaces ##V_1## of dimension 2 and ##V_2## of dimension 3. Each vector space has a basis that we...
  28. D

    I Why do the extra terms cancel in the derivation of the EM field strength tensor?

    Hi I am trying to follow the derivation in some notes I have for the field strength tensor using covariant derivatives defined by Du = ∂u - iqAu . The field strength is the defined by [ Du , Dv ] = -iqFuv The given answer is Fuv = ∂uAv - ∂vAu .When I expand the commutator I get this...
  29. O

    I Why is stress considered a tensor?

    Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor. But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations. I am searching for a more concise/abstract...
  30. F

    A Is the Metric Tensor Invariant under Lorenz Transformations in M4?

    I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...
  31. D

    I Covariant derivative of field strength tensor

    Hi, I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row) which is $$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$ doesn't match my calculation Since $$D_\rho B_{\mu\nu}=(\partial_\rho+i g...
  32. V

    I Infinitesimal cube and the stress tensor

    The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...
  33. mhob

    A Contraction between Levi-Civita symbol and Riemann tensor

    How to proof that εμνρσ Rμνρσ =0 ? Thanks.
  34. arpon

    I Is the Contraction of a Mixed Tensor Always Symmetric?

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

    Struggling immensely with tensors in multivariable calculus

    Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...
  36. 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...
  37. I

    I Maxwell's Eqs. & Tensor Notation

    In one of our lectures we wrote Maxwell's equations as (with ##c=1##) ##\partial_\mu F^{\mu \nu} = 4\pi J^\nu## ##\partial_\mu F_{\nu \rho} + \partial_\nu F_{\rho \mu} + \partial_\rho F_{\mu \nu} = 0## where the E.M. tensor is ## F^{\mu \nu} = \begin{pmatrix} 0 & -B_3 & B_2 & E_1\\ B_3 & 0 &...
  38. rezkyputra

    Covariant Derivatives (1st, 2nd) of a Scalar Field

    Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...
  39. K

    I Writing Components of a Metric Tensor

    I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this: g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu}) where g1 and g2 are functions of one variable alone and D is the...
  40. lonewolf219

    Help with tensor formulation of special relativity

    Homework Statement Hi, I can't seem to understand the following formula in my professor's lecture notes: F_αβ = g_αγ*g_βδ*F^(γδ) Homework Equations Where g_αβ is the diagonal matrix in 4 dimensions with g_00 = 1 and g_11 = g_22 = g_33 = -1 and F^(γδ) is the electromagnetic tensor with c=1...
  41. H

    How to find the Piola-Kirchhoff stress tensor

    Homework Statement Hello, I am supposed to show that the quantity TR=JTF-t satisfies TR=∂W/∂F for some scalar function W(X, F, θ) in my continuum mechanics homework. The task is to identify this scalar function W(X, F, θ).Homework Equations This is part b) of a question. In part a), we get...
  42. K

    I Is Symmetry on μ and α Valid for the Derivative of the Metric Tensor?

    I was thinking about the metric tensor. Given a metric gμν we know that it is symmetric on its two indices. If we have gμν,α (the derivative of the metric with respect to xα), is it also valid to consider symmetry on μ and α? i.e. is the identity gμν,α = gαν,μ valid?
  43. Devin

    I Stress Energy Tensor for Oscillator: Setup for Electron in E-Field

    How would one go about setting up the stress energy tensor for a particle, say an electron subjected to electric an electric field that makes the particle oscillate with frequency \omega?
  44. H

    Classical Modern Tensor Calculus/Continuum Mech Textbook

    Hi, I'm looking for a modern, colourful, illustrative introductory textbook to work through on tensor calculus/continuum mechanics. I'd like one with lots of physical examples, exercises, summaries, etc. I'd like an emphasis on engineering. Something in the mould of Frank White's Fluid...
  45. M

    I Trying to understand covariant tensor

    I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors. I looked into the Wikipedia page...
  46. C

    Understanding elastic tensor matrix intuitively

    Hi, I know the generalized hookes law between stress and strain is given by the elastic tensor. This matrix has 81 constants which are reduced to 9 in the isotropic case. Can someone please help me to understand intuitively how this reduction in the elastic tensor takes place and why some of the...
  47. K

    I Tensor networks, spin networks and loop quantum gravity

    Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy Muxin Han, Ling-Yan Hung (Submitted on 7 Oct 2016) The relation between Loop Quantum Gravity (LQG) and tensor network is explored from the perspectives of bulk-boundary duality and holographic entanglement...
  48. Eswin Paul T

    I What is a Tensor? - Get Help Here

    I have been trying to understand what a tensor is, still I cannot make an intuitive idea about it. I need help. Thanks in advance.
  49. BiGyElLoWhAt

    Moment of inertia tensor calculation and diagonalization

    Homework Statement Not sure if this is advanced, so move it wherever. A certain rigid body may be represented by three point masses: m_1 = 1 at (1,-1,-2) m_2 = 2 at (-1,1,0) m_3 = 1 at (1,1,-2) a) find the moment of inertia tensor b) diagonalize the matrix obtaining the eigenvalues and the...
  50. L

    Tensor Calculation & Lorentz Transformation: Understanding Relations

    Hi, I have trouble understanding why the following relations hold true. Given the Minkowski metric \eta_{\alpha\beta}=diag(1,-1,-1,-1) and the line segment ds^2 = dx^2+dy^2+dz^2, then how can i see that this line segment is equal to ds^2 = \eta_{\alpha\beta}dx^\alpha dx^\beta . Further, we...
Back
Top