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Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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Torsion Tensor covariant and contravariant

I want to know the relation between covariant and contravariant torsion tensors. Also please tell me that can we change the order of indices in torsion tensor?
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Converting the deformation gradient from Cartesian to Cylindrical

Suppose I have a Cartesian deformation gradient tensor F for a domain $\Omega_0$. This tensor deforms $\Omega_0$ into a new domain $\Omega_1$. Also assume that I know the values for each entry of F at ...
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How does a tensor from cotangent and tangent spaces transform?

In Sean Carroll's Spacetime and Geometry An Introduction to General Relativity Chapter 2, there is an example of tensor transformation from $x,y$ coordinates to primed ones using $$(x',y') = (\frac{2x}...
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I could not proof that curl of gradient is zero. How can I do this by using indiciant notation? [migrated]

I could not find a way to equal this to zero. $\vec{\nabla}\times(\vec{\nabla}\phi)= \epsilon_{ijk}\partial_{j}(\partial\phi)_{k}= \epsilon_{ijk}(\phi(\partial_{j}\partial_{k}) + \partial_{k}(\...
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Cannot simplify expression with rotor and nabla with index notation [on hold]

I need to handle simple operation which needs some skill in tensor algebra. I have to take $\mathrm{rot}$ from $ (\vec u \cdot\nabla)\vec u $. I am not very good at tensors operations, but I know ...
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explanation of dual tensors [closed]

I find Zee's discussion of dual tensors on pp. 192 --> of "Group Theory in a Nutshell for Physicists" incomprehensible, particularly his notation. I'd like either a link to a discussion of them or an ...
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1answer
46 views

Notation for the divergence of a rank 2 tensor

I am studying advanced fluid mechanics and sometimes you see equations written in index notation like $$ Dv_i= \partial_t v_i +v_j\partial_jv_i$$ but sometimes you find this arrow/vector notation (...
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Tensor Derivatives in Index Notation in Special Relativity

The energy-momentum tensor $T^{\mu\nu}$ is not uniquely defined because we can add a term $\partial_{\lambda}X^{\lambda\mu\nu}$ to it, where $X^{\lambda\mu\nu} = - X^{\mu\lambda\nu}$, and show that it ...
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$\nabla\cdot(\vec{a}\otimes\vec{b})$ being a row vector

I'm learning fluid physics, and I see a lot of $\nabla\cdot(\vec{a}\otimes\vec{b})$ (sometimes they just write it $\nabla\cdot(\vec{a}\vec{b})$ but I somehow don't like it) I'm proving everything in ...
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1answer
50 views

How can you use tensors in theory of elasticity? [closed]

I am interested in physics and Icame across the usage of tensors in elasticity. How do you do that? How can tensors be useful?
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1answer
25 views

Square bracket notation of the basis of 16 independent gamma matrices

The question is very simple and I couldn't find an answer. What the notation $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho ]}$ and $\gamma^{ [ \mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma ]}$ means? ...
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Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
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How to write $3\otimes3\otimes 3=10\oplus8\oplus8\oplus1$ in a tensorial way? [duplicate]

I wasn't sure how to write the title because I don't really understand this topic. Here's my question: When we are constructing hadrons we put quarks together to form higher representations of the ...
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3answers
78 views

Geodesics - Reparameterization

I am reading Wald's textbook Chapter 3. I am struggling with Section 3.3 and problem 5. It states that any curve that satisfies the weaker condition $T^{a}\nabla_{b}T^{b} = \alpha T^{a}$ is $eq.(3.3....
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42 views

Is this calculus on the Lorentz factor right?

Can the following process be justified? The Lorentz factor is given by $$\gamma (V^i)=\left(1-\frac{V^2}{c^2}\right)^{-1/2}$$ where $V^2=\sum_{i=1,2,3}{V^iV^i}$ So, the partial derivative of the ...
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With regard to distributive law of inner product in vector algebra [migrated]

Consider the equality \begin{align*} &\vec{a}\cdot\vec{b}+c=0 \\ \implies &\vec{a}\cdot(\vec{b}+\frac{\vec{a}}{\vec{a}\cdot\vec{a}}c)=0. \end{align*} If the above equation is valid for any $\...
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Zero order Tensor

One of the definition of tensor is that it should be invariant under transformation. Thus a tensor of order zero is a scalar should be invariant under transformation. For a tensor component $v^{\mu}$ ...
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104 views

Is Lorentz transform a tensor?

I am confused whether Lorentz transform is a tensor or not, since it is a linear transform. If yes how can I verify that?
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About covariant component (Dan Fleisch book)

I am familiar with the mathematical definitions of contra/co-variant components of $(p,q)$ tensors as presented in the book of smooth manifolds of John Lee and now I'm reading a more elementary book ...
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Angular velocity of rigidly rotating orbit in 3D

Consider a circle in 3-dimensional space. On this circular orbit, a rigid bead moves, thus changing its angle $\phi$ with a reference radius on the circle. The intrinsic angular velocity is given by ...
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Since the Q-criterion is a scalar, does it not change upon moving reference frames?

Based on the definition from An objective definition of a vortex, the $Q$-criterion is $$Q=\frac{1}{2}(|\boldsymbol{\Omega}|^2-|\mathbf{S}|^2)$$ where $\boldsymbol{\Omega}$ is the rotation rate tensor,...
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Question about Ricci Rotation Coeficients

Standard General Relativity calculations lies under, indeed, the calculations of three quantities: Christoffel Symbols of second kind, the components of Riemann tensor $R^{\mu}\hspace{1mm}_{\nu \gamma ...
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Action of a 1-form on the push-forward and pull-back of a vector [migrated]

I am studying differential geometry I am trying to proof the expression below. Given that for a map $\phi$ : $M$ $\to$ $M$ the pull-back $\phi$*$\omega$ $\in$ $T^\ast_p M$ of a 1-form $\omega $ $ \...
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What are some of the reasons for raising/lowering indices of a tensor?

In Dirac's paper: Classical theory of radiating electrons, he decides to raise and lower the indices on the same object multiple times: \begin{align*} \frac{\partial{A_{\mu}}}{\partial{x_{\mu}}} &...
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1answer
62 views

Representing a reducible Cartesian tensor as a spherical tensor

I'm quite confused by this transformation, and am trying to gain fluency in moving back and forth between these objects. I understand that a second order dyadic Cartesian tensor can be represented as ...
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1answer
38 views

Is there a simplification to the triple product of levi-civita symbols

IS there any way to simplify: $\epsilon_{ijk}\epsilon_{klm}\epsilon_{mni}$ I know how to simplify the product of just two into summations of delta functions but not sure how to do it with three
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1answer
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Need some help with a question involving the Levi-Civita symbol [closed]

Been struggling with this question: Given: $c_{ij}=\epsilon_{ijk}a_k$ I have to show that $a_k=\frac{1}{2}\epsilon_{kij}c_{ij}.$ I was thinking of multiplying both sides by some epsilon, but not ...
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How can the following tensor can be reexpressed by totally symmetrized derivatives? See the attached picture

Particularly, in Wald paper https://doi.org/10.1063/1.528839 from tensor in (4) to (5).
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1answer
51 views

How to use tensors and operators

I have some problem understanding how to use tensors. Let's say in Quantum Optics if I have the state in mode $b$ (where I can have two possible modes $a$ and $b$) $$|1_b\rangle = |0_a\rangle \otimes|...
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2answers
53 views

Stress tensor of an elastic medium

I don't understand a passage from the book I'm reading about tensor analysis. The state of stress of an elastic medium can be expressed by the stress function $\mathbf{p}(r,n)$ so that the force ...
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2answers
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How does 4-vector notation work?

In particle physics we are going over 4-vector notation. However, my background on this is a little shaky, and I'm having difficulty differentiating the notation and visualizing what it actually means....
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Maxwell-Chern-Simons equation: Translating from differential form to component form

I am trying to solve the scalar-coupled Maxwell-CS equations (which is one of the equation of motions in $N=2$ supergravity coupled to 3 vector multiplets), which is written in this form in the ...
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1answer
50 views

Naive question about tensor densities

Well, a (contravariant) tensor is an element of tensor product: $$V\otimes W =: \frac{F(V\times W)}{S}$$ as exposed here: https://textosdefisica.wordpress.com/textos-de-matematica/ A antisymmetric ...
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1answer
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Instructions for mapping the independent Riemann coefficients to the redundant Riemann coefficients

Introduction: I have been developing a General Relativity utility for working out the stress tensor coefficients for a given metric and all the related Riemannian coefficients which build up to it: ...
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1answer
70 views

Index position when varying an action with respect to the metric

I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as $$ A_{\mu\nu}B^{\mu\nu}, $$ do I necessarily have ...
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1answer
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Differentiating Scalar along a geodesic

I have been studying GR for sometime and doing exercises from Schutz and I have a question about differentiating along a geodesic. Here is what I know. The equation of geodesic in terms of four ...
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5answers
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Could be possible to build a 4-vector in special relativity whose spatial component was the electric field E?

Hi everyone and sorry for my English. I would like to know if I can build a legitimate 4-vector as $E^\alpha=(E^0,\mathbf{E})$. I'd like you to check if my way is correct. 1- We already know that $\...
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Question about tensor integration

A (covariant)$(0,2)$-tensor can be written as: $$T = \sum_{\mu}\sum_{\nu}t_{\mu \nu} e^{\mu}\bar{\otimes}e^{\nu}\tag{1}$$ with the particular basis vectors $e^{\mu}\bar{\otimes}e^{\nu}$ that spans ...
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1answer
70 views

Line element 1-form

It was pointed out that dual vectors of a manifold, and hence differential 1-forms, are not dependent on the metric (Intuition behind dual vectors ('Bongs of a bell' does not help)). But doesn'...
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3answers
102 views

Intuition behind dual vectors ('Bongs of a bell' does not help)

Similar to the post here (How to visualize the gradient as a one-form?), I'm wondering about an intuition behind dual vectors and differential forms (and the link in that answer to Thorne's notes is ...
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Momentum in capacitor field; How can an EM field have zero momentum density but non-zero momentum flux?

Consider the case of a simple, stationary parallel plate capacitor oriented with its plates lying in the x-y plane. The E-field is simply given by: $$\vec{E} = \frac{Q}{\epsilon_0A}\hat{z} $$ with ...
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2answers
189 views

Doubts on covariant and contravariant vectors and on double tensors

I'm trying to study tensors. Given a coordinates transformation from cartesian to $u_i$ ones: $$ u_1 = u_1 (x,y,z) \qquad u_2 = u_2 (x,y,z) \qquad u_3 = u_3 (x,y,z) $$ I can write a vector $\mathbf{...
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2answers
59 views

4-Gradient vector and the Field strength tensor

Need some help evaluating the following 4-gradient, that of the gradient of the field strength tensor $$F^{\mu\nu}= \begin{bmatrix} 0 & -E_x & -E_y & -E_z\\\ E_x & 0 & -...
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1answer
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How to construct a 2-partite matrix

Let's assume we have an internal hamiltonian $H_0 = \mid 1\rangle \langle 1\mid$. Now let's assume we have two systems with identical Hamiltonians $H^1_0$,$H^2_0$ and I want to compute the joint ...
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2answers
117 views

Is there a useful way to visualize the symmetries of the relativistic Riemann curvature tensor?

I find it useful to see diagrams such as trees, colored 2D and 3D arrays, etc., which illustrate how terms combine in composite expressions. For example, the following is my visualization of the ...
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1answer
63 views

Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
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1answer
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MTW Box 4.1 Contraction of EM 2-form with surface element bivector to give “magnetic flux”. What does that mean?

This is my attempt to make sense of Box 4.1-4.b of MTW's Gravitation. I'm not entirely sure I have the computation correct. But, even if it is correct, I don't really understand how the final result ...
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1answer
64 views

When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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2answers
87 views

Is there a good treatment of “familiar” physics using exterior calculus, AKA differential forms?

By familiar physics, I mean the physics of things I can reach out and touch. In other words, neither relativity nor analytical dynamics, etc. After re-reading chapter 4 of MTW's Gravitation yet ...
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1answer
56 views

(Lorentz etc) invariant vector fields

(Background: I know some but not much differential geometry, hopefully enough to formulate this post.) I want to ask about what physicists mean when they say scalar, vector, etc. The answer in ...