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. S

    Density terms in the stress-energy momentum tensor

    The stress energy momentum tensor of the Einstein field equations contains multiple density terms such as the energy density and the momentum density. I know how to calculate relativistic energy and momentum, but none of the websites or videos that I have watched make mention of any division of...
  2. S

    Understanding the Ricci Curvature Tensor in Einstein's Field Equations

    I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...
  3. T

    Why Use Tensors in GR: Benefits & Potential Pitfalls

    I think that is a fundamental question of why we need Tensor when dealing with GR? Quoting from the textbook (Relativity, Gravitation and Cosmology: A Basic Introduction) Tensors are mathematical object having definite transformation properties under coordinate transformations. The simplest...
  4. J

    Is the Modulus of a Tensor Calculated Differently Than a Vector?

    I was thinking... if the modulus of a vector can be calculated by ##\sqrt{\vec{v} \cdot \vec{v}}##, thus the modulus of a tensor (of rank 2) wouldn't be ##\sqrt{\mathbf{T}:\mathbf{T}}## ?
  5. P

    Parity conservation and the Field-Strength Tensor‏

    In reexamining chapter 11 of Jackson's Classical Electrodynamics, especially equations 11.148, it seems obvious that in placing the E and B transformation values into the electro-magnetic field-strength tensor one is ignoring the standard rules which do not allow combining polar vectors and...
  6. Math Amateur

    MHB Tensor Products - Example 8 - Dummit and Foote - Section 10.4, page 370

    I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example (8) on page 366 concerning viewing the quotient ring R/I as an (R/I, R) -bimodule. Example (8) D&F page 370 reads as follows: (see attachment)...
  7. Math Amateur

    MHB Understanding D&F Example 2: R/I Bimodule on Page 366

    I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example 2 on page 366 concerning viewing the quotient ring R/I as an (R/I, R) -bimodule. Example (2) D&F page 366 reads as follows...
  8. A

    Nuclear force tensor operator expectation value.

    Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...
  9. T

    Understanding Stress Tensor in MTW Ex. 5.4

    I've been working on Ex 5.4 in MTW. The maths is fairly straight forward, but I don't really understand what is going on! In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these...
  10. Math Amateur

    MHB Simple Problem concerning tensor products

    Actually this problem really only concerns greatest common denominators. In Section 10.4, Example 3 (see attachment) , Dummit and Foote where we are dealing with the tensor product \mathbb{Z} / m \mathbb{Z} \otimes \mathbb{Z} / n \mathbb{Z} we find the following statement: (NOTE: d is the gcd...
  11. Math Amateur

    MHB Tensor Products - D&F page 369 Example 3 - The map phi

    I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: (see attachment) ------------------------------------------------------------------------------- In general...
  12. Math Amateur

    MHB Tensor Products - D&F page 369 Example 2

    I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: ------------------------------------------------------------------------------- In general, \mathbb{Z} / m...
  13. C

    Stress-energy tensor explicitly in terms of the metric tensor

    I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...
  14. Math Amateur

    MHB Tensor Products - Dummit and Foote Section 10.4, Example 2, page 363

    I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I am currently studying Example 2, page 363 (see attachment) and I am trying to closely relate the example to Theorem 8 and the D&F text on extension of the scalars preceding Theorem 8 on pages 359-362) In Example 2 (see...
  15. Math Amateur

    MHB Tensor Products - Dummit and Foote Section 10.4 Corollary 9

    I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am studying Corollary 9 and attempting to fully understand the Corollary and it proof. (For details see the attachement page 362 in which Theorem 8 is stated and proved. This is followed by the statement and proof of...
  16. Math Amateur

    MHB Is There Only One Possible Z-Linear Map from T to T'?

    First, thanks to both Deveno and ThePerfectHacker for helping me to gain a basic understanding of tensor products of modules. In a chat room discussion ThePerfectHacker suggested I show that {\mathbb{Z}}_a \otimes_\mathbb{Z} {\mathbb{Z}}_b where a and b are relatively prime integers - that is...
  17. Math Amateur

    MHB Tensor Products - The free Z-module construction

    I am trying (struggling! :() to understand tensor products as developed by Dummit and Foote in Section 10.4 - specifically the early section devoted to the "extension of scalars". I have been reflecting on my attempts to understand the material of Dummit and Foote, pages 359 -362 (see...
  18. Math Amateur

    MHB Tensor Products - Dummit and Foote - Section 10-4, Theorem 8, page 362

    I am reading Dummit and Foote, Section 10 on tensor products of modules. I am at present trying to understand the use of Theorem 6 (D&F, page 354 - see attachment) in Theorem 8 (D&F page 362, see attachment). The proof of Theorem 8 in D&F Chapter 10 (see attachment) reads as follows...
  19. Math Amateur

    MHB Tensor Products - D&F - Extension of the scalars

    I am attempting to understand Dummit and Foote exposition on 'extending the scalars' in Section 10.4 Tensor Products of scalars - see attachment - particularly page 360) [I apologise in advance to MHB members if my analysis and questions are not clear - I am struggling with tensor products! -...
  20. C

    Solving Newtonian Tensor from A. Zee's EGR Book

    I'm working through A. Zee's new EGR book, and I came to a step on tidal forces I couldn't follow. He presents the gravitational potential V(\vec{x})=-GM/r and asks us to verify that the tensor R^{ij}(\vec{x})\equiv\partial^{i}\partial^{j}V(\vec{x}) is, in this case...
  21. C

    Transformation Properties of a tensor

    Homework Statement ##D_{ijk}## is an array with ##3^3## elements, which is not known to represent a tensor. If for every symmetric tensor represented by ##a_{jk}## $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the...
  22. C

    Is the metric tensor constant in polar coordinates?

    I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by: g^{\mu \nu} = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix} Since...
  23. Math Amateur

    MHB Tensor Products - Dummit and Foote - Section 10-4, pages 359 - 362

    In Dummit and Foote, Section 10.4: Tensor Products of Modules, on pages 359 - 364 (see attachment) the authors deal with a process of 'extension of scalars' of a module, whereby we construct a left S-module S \oplus_R N from an R-module N. In this construction the unital ring R is a subring of...
  24. Math Amateur

    MHB Tensor Products - Keith Conrad - Theorem 3.3 - Tensor Products I

    I am reading and trying to fully understand Keith Conrad's paper: Tensor Products I. These notes are available at Expository papers by K. Conrad or the specific paper at http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf. Conrad's Theorem 3.3 (see attachment - page 10) is...
  25. D

    Why is the stress-energy tensor symmetric?

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  26. Math Amateur

    MHB Existence of Tensor Products - Keith Conrad - Tensor Products I - Theorem 3.2

    I am reading and trying to follow the notes of Keith Conrad on Tensor products, specifically his notes: Tensor Products I (see attachment ... for the full set of notes see Expository papers by K. Conrad ). I would appreciate some help with Theorem 3.2 which reads as follows: (see attachment...
  27. M

    Y^2 - x^2 in the [itex]\mid n\ell m \rangle[/itex] basis - tensor Op.

    x^2 - y^2 in the \mid n\ell m \rangle basis - tensor op. Homework Statement I must determine the matrix elements of x^2 - y^2 in the \mid n\ell m \rangle basis. "...use the fact that x^2 - y^2 is a sum of spherical components of a rank two tensor, together with the explicit form of the...
  28. Math Amateur

    MHB Introduction to Tensor Products - some advice please

    I am (trying to :-) ) reading Dummit and Foote Section 10.4 on Tensor Products of Modules and am finding D&F's introduction to the topic of tensor products quite bewildering! ... Can anyone give me a simple definition of a tensor product of modules together with an example to give me a basic...
  29. P

    Attempting to learn tensor calculus

    Hello all, After a brief break from attempting to learn tensor calculus, I'm once again back at it. Today, I started reading this: http://web.mit.edu/edbert/GR/gr1.pdf. I got to about page 4 before things stopped making sense, right under equation 3. Question 1: apparently a "one-form" is a...
  30. M

    Understanding Viscous Stress Tensor in Incompressible Flow

    hey pf! in reading a book on viscous stresses i found the following: \tau_{ij}=2\mu\Big(s_{ij}-\frac{1}{3}s_{kk}\delta_{ij}\Big) where einstein summation is used. now we have s_{ij}=\frac{1}{2}\Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\Big) and then the claim is...
  31. D

    Tensor Densities: Coordinate Independent Definition

    Is there a coordinate independent/geometric definition of a tensor density?
  32. A

    Field strength tensor / matrix

    In my note, we have written the field strength tensor as: F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = \begin{pmatrix} 0&E_x &E_y&E_z \\ -E_x&0 &B_z &-B_y \\ -E_y&-B_z &0 &B_x \\ -E_z&B_y &-B_x&0 \end{pmatrix} But if I look into...
  33. S

    Stress-energy tensor for a single photon

    Hi, I'm trying to write down the stress-energy tensor for a single photon in GR, but I'm running into trouble with its transformation properties. I'll demonstrate what I do quickly and then illustrate the problem. Given a photon with wavevector p, we write {\bf T} = \int \frac{\mathrm{d}^3...
  34. andrewkirk

    Intuitive description of what the Ricci tensor & scalar represent?

    Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined...
  35. S

    How do I convert an inertia tensor from body space to world space?

    I've been trying to solve the following problem involving Angular Acceleration and the inertial tensor for about 2 weeks now. I know it's bad ask for a question to be solved, but I'm really at a loss here folks. I'm a high school student who has taken a physics class. What I'm Trying To Do...
  36. N

    Finding inverse metric tensor when there are off-diagonal terms

    How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...
  37. D

    Tensor Rank vs Type: Explained

    Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2...
  38. J

    Scalar, vector and tensor calculus

    I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...
  39. bcrowell

    Maxwell's equations from divergence of stress-energy tensor?

    If I start with the stress-energy tensor T^{\mu\nu} of the electromagnetic field and then apply energy-momentum conservation \partial_\mu T^{\mu\nu}=0, I get a whole bunch of messy stuff, but, e.g., with \nu=x part of it looks like -E_x \nabla\cdot E, which would vanish according to Maxwell's...
  40. D

    What is the Lorentz invariance of flux in the stress-energy tensor?

    According to Wikipedia, This definition doesn't sit well with me. Flux is defined as the rate that something passes through an infinitesimal surface, divided by the infinitesimal area of that surface. For example, the current flux (or current density), when dotted with a unit vector, gives...
  41. C

    Metric tensor after constructing a quotient space.

    Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...
  42. S

    The stifle-Whitney classes of a tensor product

    Homework Statement Let ξm and ηn be vector bundles over a paracompact base space. Show that the stifle-Whitney classes of the tensor product ξm ⊗ ηn (or of the isomorphic bundle Hom (ξm, ηn) can be computed as follows. If the fiber dimensions m and n are both 1 then: w1 (ξ1 ⊗ η1) = w1(ξ1) +...
  43. L

    Evaluating if a Vector is a Tensor

    Homework Statement Problem as stated: Consider a vector A^a. Is the four-component object \left( \frac{1}{A^0},\frac{1}{A^1},\frac{1}{A^2},\frac{1}{A^3}\right) a tensor? Homework Equations Roman indices run from 0 to 3. Einstein summation convention is used. Tensors of rank 1 (vectors)...
  44. T

    Stress tensor transformation and coordinate system rotation

    Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...
  45. P

    Stress energy tensor for fields

    In the case of swarms of particles, the stress energy tensor can be derived by considering the flow of energy and momentum "carried" by the particles along their worldlines. Is there a way to interpret the field definition of the stress energy tensor from Wald, p455 E.1.26 T_{ab} \propto...
  46. M

    The Belinfante_rosenfeld tensor

    Hi guys, Can anyone please help me to grasp a minor detail in the derivation of the Belinfante-Rosenfeld version of the Stress-Energy Tensor (SET) ? To save type, I refer to the wiki webpage http://en.wikipedia.org/wiki/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor Using...
  47. N

    Index Notation for Rank-2 Tensor with Summation

    Homework Statement I have the following rank-2 tensor T = \nabla \cdot \sum_{i}{c_ic_ic_i} I would like to write this using index notation. According to my book it becomes T_{ab} = \partial_y \sum_{i}{c_{ia}c_{ib}c_{iy}} Question: The change \nabla \rightarrow \partial_y and c_i...
  48. facenian

    Is the Linear Eulerian Deformation Tensor an Exact Measure of Deformation?

    Hello, I have difficulty interpreting the following fact (I'm reading Cotinuum Mechanics by Spencer). The relative velocity between two nearby points P and Q in the current configuarion is given by: dv_i=D_{ik}dx_k + W_{ik}dx_k where D_{ik}=\frac{d}{dt}e_{ik} is the rate of deformation tensor...
  49. M

    Transformation of the metric tensor from polar to cartesian coords

    I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...
  50. N

    Divergence of a rank-2 tensor in Einstein summation

    Homework Statement Hi When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} +...
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