tensor Definition and Topics - 92 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

    I Proving ##\partial^{i} = g^{ik} \partial_{k}##

    Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that $$ \frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k} $$ I wanted to prove this identity. Using the chain rule $$ \frac{\partial}{\partial x_{i}}=\frac{\partial...
  2. A

    I Parallel transport of a tensor

    According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is: ##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0## Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
  3. A

    I Derivation of the contravariant form for the Levi-Civita tensor

    The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##. My attempt What I have tried is to express this...
  4. A

    I Showing that the determinant of the metric tensor is a tensor density

    I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
  5. A

    I Doubt about the purpose of some elements in tensor calculus

    I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it. Tensors. As...
  6. T

    Help with Kaluza Klein Christoffel symbols

    If I want to calculate ##\tilde{\Gamma}^\lambda_{\mu 5}##, I will write \begin{align} \tilde{\Gamma}^\lambda_{\mu 5} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{5 X} + \partial_5 \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu 5}\right) \\ & =\frac{1}{2}...
  7. C

    Calculation Involving Projection Tensor in Minkowski Spacetime

    In Minkowski spacetime, calculate ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}##. I had calculated previously that ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}## When I subsitute it back into the expression...
  8. T

    Trying to understand electric and magnetic fields as 4-vectors

    I was trying to show that the field transformation equations do hold when considering electric and magnetic fields as 4-vectors. To start off, I obtained the temporal and spatial components of ##E^{\alpha}## and ##B^{\alpha}##. The expressions are obtained from the following equations...
  9. T

    Tensor help -- Write out this tensor in a simplified sum

    I managed to write $$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$ where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$. How do I proceed?
  10. L

    I Permittivity as a tensor

    If I have an anisotropic material with permittivity: $$\epsilon= \begin{pmatrix} \epsilon_{ii} & \epsilon_{ij} & \epsilon_{ik} \\ \epsilon_{ji} & \epsilon_{jj} & \epsilon_{jk} \\ \epsilon_{ki} & \epsilon_{kj} & \epsilon_{kk} \\ \end{pmatrix} $$ What exactly does each element represent in this...
  11. George Keeling

    I Coordinate conversion in Orodruin's Insight

    I am studying @Orodruin's Insight "Explore Coordinate Dependent Statements in an Expanding Universe". It looks pretty interesting. About three pages in it reads "expanding ##x^a## to second order in ##\xi^\mu## generally leads to$$ x^a=e_\mu^a\xi^\mu+c_{\mu\nu}^a\xi^\mu\xi^\nu+\mathcal{O}_3...
  12. George Keeling

    I Tensor rank: One number or two?

    When I started learning about tensors the tensor rank was drilled into me. "A tensor rank ##\left(m,n\right)## has ##m## up indices and ##n## down indices." So a rank (1,1) tensor is written ##A_\nu^\mu,A_{\ \ \nu}^\mu## or is that ##A_\nu^{\ \ \ \mu}##? Tensor coefficients change when the...
  13. F

    Find the tensor that carries out a transformation

    I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##. ##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}## ##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
  14. K

    I Prove that dim(V⊗W)=(dim V)(dim W)

    This proof was in my book. Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively. I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
  15. P

    I Irreducible representations of the EM-tensor under spatial rotations

    Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...
  16. K

    Calculate the angle between the E-field and Current vectors in an anisotropic conductive material

    In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...
  17. K

    Show that a (1,2)-tensor is a linear function

    I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
  18. fabstr1

    Simplification of the Proca Lagrangian

    Hello, I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it. ∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1) L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2) Here is Eq (1) the...
  19. LuccaP4

    I 4th-rank isotropic tensor

    I have this statement: Find the most general form of the fourth rank isotropic tensor. In order to do so: - Perform rotations in ## \pi ## around any of the axes. Note that to maintain isotropy conditions some elements must necessarily be null. - Using rotations in ## \pi / 2 ## analyze the...
  20. K

    A Differential Forms or Tensors for Theoretical Physics Today

    There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
  21. cianfa72

    I About Covariant Derivative as a tensor

    Hi, I've been watching lectures from XylyXylyX on YouTube. I believe they are really great ! One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
  22. W

    A A doubt about an integral along a null geodesic

    I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$ And I apply the worm hole space...
  23. Vyrkk

    A Covariant derivative and connection of a covector field

    I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors. What people usually do is take the covariant derivative of the covector acting on a vector, the result being a scalar Invoke a product rule to...
  24. A

    I Meaning of each member being a unit vector

    Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...
  25. nomadreid

    I Tensor calculation, giving|cos A|>1: how to interpret

    On pages 42-43 of the book "Tensors: Mathematics of Differential Geometry and Relativity" by Zafar Ahsan (Delhi, 2018), the calculation for the angle between Ai=(1,0,0,0) (the superscript being tensor, not exponent, notation) and Bi=(√2,0,0,(√3)/c), where c is the speed of light, in the...
  26. W

    Metric tensor problem

    My attempt at ##g_{\mu \nu}## for (2) was \begin{pmatrix} -(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta) \end{pmatrix} and the inverse is the reciprocal of the diagonal elements. For (1) however, I can't even think of how to write the...
  27. D

    I Deriving tensor transformation laws

    Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc. I'm going to define a tensor. $$T^\mu_\nu \partial_\mu \otimes dx^\nu$$ Now I'd like to investigate how the tensor transforms...
  28. sergiokapone

    A Index juggling by example of angular momentum tensor

    Lets consider the angular momentum tensor (here ##m=1##) \begin{equation} L^{ij} = x^iv^j - x^jv^i \end{equation} and rortational velocity of particle (expressed via angular momentum tensor) \begin{equation} v^j = \omega^{jm}x_m. \end{equation} Then \begin{equation} L^{ij} =...
  29. A

    I Tensor product expansion

    Hi, I'm currently working through a tensor product example for a two qubit system. For the expression: $$ \rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle $$ Which has been defined as from going to a global state to a local state. Here $$ |\Psi \rangle = |\Psi^+...
  30. M

    A Meaning of the Riemman Tensor notation in the choquet Bruhat

    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...