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. W

    Short Exact Sequences and at Tensor Product

    Hi,let: 0->A-> B -> 0 ; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B. . We have that tensor product is right-exact , so that, for a ring R: 0-> A(x)R-> B(x)R ->0 is also exact. STILL: are A(x)R , B(x)R isomorphic? I suspect no, if R has torsion. Anyone...
  2. Mr-R

    I Calculating the Riemann Tensor for a 4D Sphere

    Dear All, I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed} But the exercise asks me to calculate R_{abcd}. Do...
  3. S

    Stress Energy Tensor Components

    I have pretty much learned how to derive the left side of Einstein's field equations now (the Einstein tensor that is). Now I need to grasp that stress energy momentum tensor. Does anybody know of any good sources that will tell me how to derive the components of this tensor? I ask this...
  4. G

    What Are the Best Books for Mastering Tensor Calculus?

    Hi guys, I am interested to learn tensor calculus but I can't find a good book that provide rigorous treatment to tensor calculus if anyone could recommend me to one I would be very pleased.
  5. C

    Constructing the electromagnetic tensor from a four-potential

    *Edit: I noticed I may have posted this question on the wrong forum... if this is the case, could you please move it for me instead of deleting? thanks! :) Hello, I am having problems on building my electromagnetic tensor from a four-potential. I suspect my calculations are not right. Here are...
  6. H

    What Does the Definition of a Contravariant Vector Mean in Tensor Analysis?

    So I'm looking at Schaum's outlines for Tensors and the definition of a Contravariant vector is \bar{T}^i=T^r\frac{\partial\bar{x}^i}{\partial x^r} Where \bar{x}^i and x^r denote components of 2 different coordinates (the superscript does not mean 'to the power of') and T^i and T^r are...
  7. S

    Metric Tensor in Spherical Coordinates

    I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...
  8. A

    Is stress tensor symmetric in Navier-Stokes Equation?

    Hello, In CFD computation of the Navier-Stokes Equation, is stress tensor assumed to be symmetric? We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then...
  9. D

    Curl in 5D using levi-civita tensor

    i really lost with this. i see two possibilities: (1) something like, \epsilon_{abc}\partial_{a}A_{b}e_{c} with a,b,c between 1 and 5 or (2)like that \epsilon_{abcde}\partial_{a}A_{b} one of the options nears correct? thank's a lot
  10. electricspit

    Why Does Applying a Second Derivative to an Antisymmetric Tensor Yield Zero?

    Hello, I have two problems. I'm going through the Classical Theory of Fields by Landau/Lifshitz and in Section 32 they're deriving the energy-momentum tensor for a general field. We started with a generalized action (in 4 dimensions) and ended up with the definition of a tensor...
  11. bcrowell

    Parity of stress tensor versus stress-energy tensor

    The stress-energy tensor is an actual tensor, i.e., under a spacetime parity transformation it stays the same, which is what a tensor with two indices is supposed to do according to the tensor transformation law. This also makes sense because in the Einstein field equations, the stress-energy...
  12. G

    Can Tensor Products Define (M,N) Tensors?

    From my understanding, an arbitrary (0,N) tensor can be expressed in terms of its components and the tensor products of N basis one-forms. Similarly, an arbitrary (M,0) tensor can be expressed in terms of its components and tensor products of its M basis vectors. What about an (M,N) tensor...
  13. T

    Finding rate of change of moment of inertia tensor

    Homework Statement The Wikipedia article on spatial rigid body dynamics derives the equation of motion \boldsymbol\tau = [I]\boldsymbol\alpha + \boldsymbol\omega\times[I]\boldsymbol\omega from \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i). But, there is another way to...
  14. T

    Tensor differentiation (element-by-element)

    Homework Statement Proof the following: \frac{\text{d}\boldsymbol\{\mathbf{I}\boldsymbol\}}{\text{d}t} \, \boldsymbol\omega = \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega) where \boldsymbol\{\mathbf{I}\boldsymbol\} is a tensor...
  15. Dale

    Expansion tensor on rotating disk

    Hi Everyone, Suppose that we have cylindrical coordinates on flat spacetime (in units where c=1): ##ds^2 = -dt^2 + dr^2 + r^2 d\theta^2 + dz^2## I would like to explicitly calculate the expansion tensor for a disk of constant radius R<1 and non-constant angular velocity ##\omega(t)<1##. I...
  16. C

    What kind of isometry? A metric tensor "respects" the foliation?

    Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...
  17. W

    Defining Functions on Tensor Products

    Hi all. This question is related to my previous one on tensor products: Is there a way of "well-defining" a function on a tensor product M(x)N (where M,N are both R-modules) ? This is the motivating example for my question : Say we want to define a map f: M(x)M-->M by f(m(x)m')=m+m' ...
  18. W

    Tensor Products and Maps Factoring Through

    Hi, I understand the tensor product of modules as a new module in which every bilinear map becomes a linear map. But now I am trying to see the Tensor product of modules from the perspective of maps factoring through, i.e., from properties that allow a commutative triangle of maps. As a...
  19. C

    Understanding tensor operators

    The definition of tensor operator that I have is the following: 'A tensor operator is an operator that transforms under an irreducible representation of a group ##G##. Let ##\rho(g)## be a representation on the vector space under consideration then ##T_{m_c}^{c}## is a tensor operator in the...
  20. C

    Ricci Tensor Proportional to Divergence of Christoffel Symbol?

    I'm reading an old article published by Kaluza "On the Unity Problem of Physics" where i encounter an expression for the Ricci tensor given by $$R_{\mu \nu} = \Gamma^\rho_{\ \mu \nu, \rho}$$ where he has used the weak field approximation ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}## where...
  21. F

    Energy-momentum tensor, lagrangian density

    Homework Statement I try to calculate the energy tensor, but i can't do it like the article, and i don't know, i have a photo but it don't look very good, sorry for my english, i have a problem with a sign in the result Homework Equations The Attempt at a Solution In the photos...
  22. K

    Questions about tensor operator

    Hi. Before question, sorry about my bad english. It's not my mother tongue. My QM textbook(Schiff) adopt J x J = i(h bar)J. as the defining equations for the rotation group generators in the general case. My question is, then tensor J must have one index which has three component? (e.g...
  23. E

    Why shear stress components of the Stress Energy tensor not zero?

    Hi, I am having trouble understanding why Tij can be non-zero for i≠j. Tij is the flux of the i-th component of momentum across a surface of constant xj. Isn't the i-th component of momentum tangent to the surface of constant xj and therefore its flux across that surface zero? What am I...
  24. ChrisVer

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  25. W

    What are the independent terms in the Magnetic Tensor

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  26. TrickyDicky

    Vector fields, flows and tensor fields

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  27. B

    Need clarification on the product of the metric and Levi-Civita tensor

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  28. F

    Quantitative Meaning of Ricci Tensor

    Hello, I am studying general relativity right now and I am very curious about the Ricci tensor and its meaning. I keep running into definitions that explain how the Ricci tensor describes the deviation in volume as a space is affected by gravity. However, I have yet to find any quantitative...
  29. A

    Understanding Inertia Tensor Scaling in CAD Models - Explained

    Hi everyone, I have the following problem in my hands, which I don't know how exactly to address. Let's assume that from any CAD(Solidworks, Catia), I obtain the inertia tensor of my model (impossible to calculate by hand btw). I_full=[Ixx Ixy Ixz Ixy Iyy Iyz Ixz Iyz Izz] I...
  30. J

    Rate-of-strain tensor in cylindrical coords.

    Hi PF, I posted this in HW a week ago and got no response. Might be a bit beyond the typical HW forum troller. So, please excuse the double-post. Homework Statement I'm trying to derive the rate-of-strain tensor in cylindrical coords, starting with the Christoffel symbols. Homework...
  31. D

    Is there any difference between Metric, Metric Tensor, Distance Func?

    From what I've understood, 1) the metric is a bilinear form on a space 2) the metric tensor is basically the same thing Is this correct? If so, how is the metric related to/different from the distance function in that space? Some other sources state that the metric is defined as the...
  32. ChrisVer

    Preliminary knowledge on tensor analysis

    I am not sure whether this needs to be transported in another topic (as academic guidance). I have some preliminary knowledge on tensor analysis, which helps me being more confident with indices notation etc... Also I'm accustomed to the definition of tensors, which tells us that a tensor is an...
  33. nomadreid

    Confusion about basis vectors and matrix tensor

    In "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, I read that the covariant metric tensor gij=ei°ei (I'm leaving out the → s above the e's) where ei and ei are coordinate basis vectors and ° denotes the inner product, and similarly for the contravariant metric tensor using dual...
  34. O

    Symmetrization of a tensor in spherical coordinate

    Hello, i don't know if my question is well posed, if i have a symmetric tensor Sij = (∂ixj + ∂jxi) / 2 with xi cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,\varphi)? (I need it for the calculus of shear stress tensor in spherical coordinate in fluid...
  35. W

    Calculating the Inertia Tensor of cone with uniform density

    Homework Statement Calculate the moments of inertia I_1, I_2, and I_3 for a homogeneous cone of mass M whose height is h and whose base has a radius R. Choose the x_3 axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone, and calculate the elements of the...
  36. M

    Why Does the Stress-Energy Tensor Conservation Lead to a Surface Integral?

    Homework Statement Hello I'm trying to self study A First Course in General Relativity (2E) by Schutz and I've come across a problem that I need some advice on. Here it is: Use the identity Tμ\nu,\nu=0 to prove the following results for a bounded system (ie. a system for which Tμ\nu=0...
  37. Z

    Field Strenght Tensor and its Dual (in SR)

    Hello everyone, I have recently read a puzzling statement on my Electromagnetism (Chapter on Special Relativity) material regarding the Field Strength Tensor, F^{\mu\nu}, and its dual, \tilde{F}^{\mu\nu}. Since I've been thinking about this for a while now, and still can't understand it, I...
  38. S

    Maxwell Stress Tensor -> Force between magnets and perfect iron

    (this is not a hw) Assume you have a magnet of dimensions x_m, h_m, remanent flux density Br, and coercive field density Hc. The magnet is placed in a magnetic "C" structure (perfect iron) such that it is connected on one side but there is an airgap on the other side. xxxxxxxx xx... xx...
  39. W

    Conserved Charges of Stress Energy Tensor

    Hello, Hi There I am trying to obtain the relations of the conserved charges of the stress tensor, it has 4, one is the hamiltonian and the other three are the momentum components. \vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))} And i have to prove the conmutators...
  40. W

    What are independent terms in Magnetic Tensor

    I am trying to understand the magnetic gradient tensor which has nine components. There are three magnetic field components, but there are also three baselines. These nine gradients are organised into a 3x3 matrix. I have read that only 5 of these terms are independent. What exactly does this...
  41. jk22

    Use of component tensor product in quantum mechanics?

    suppose we consider the measurement operator A=diag(1,-1). Then the tensor product of A by itself is in components : A\otimes A=a_{ij}a_{kl}=c_{ijkl} giving c_{1111}=c_{2222}=1, c_{1122}=c_{2211}=-1 and all other component 0. to diagonalize a tensor of order 4, we write ...
  42. shounakbhatta

    Riemann Metric Tensor: Exploring Basics

    Hello All, Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...
  43. P

    Tensor of inertia - hollow cube.

    Hi, Homework Statement I have found the tensor of inertia of a rectangle of sides a and b and mass m, around its center, to be I11=ma2/12, I22=mb2/12, I33=(ma2 + mb2)/12. All other elements of that tensor are equal to zero. I would now like to use this result to determine the tensor of inertia...
  44. Sudharaka

    MHB Difference Between Tensor Product and Outer Product

    Hi everyone, :) Xristos Lymperopoulos on Facebook writes (>>link<<);
  45. H

    What is the Tensor Product and its Properties in Different States?

    Dear All, I need some explanations of properties of tensor and the tensor product on different states; σ1ijσij2=_____________ Thank you.
  46. D

    Help understanding minkowski tensor and indices

    So I have just been introduced to indices, four vectors and tensors in SR and I'm having trouble knowing exactly what I am being asked in some questions. So the first question asks to write explicitly how a covariant two tensor transforms under a lorentz boost. Now I know that it transforms...
  47. 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...
  48. 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...
  49. 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...
  50. 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}}## ?
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