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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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Show the density is an objective quantity

I am studying the Lagrangian and Eulerian representations of quantities within the scope of fluid dynamics. At some point I am asked to show that the density is an objective quantity, which for a ...
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1answer
29 views

Isotropy of the Minkowski metric

An isotropic tensor has the same components in all rotated coordinate systems. Scalars or tensors of rank zero are isotropic. Tensors of rank one or vectors are not isotropic. The only isotropic rank-...
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Proof of invariance of scalar product under rotations, using index notation

So I have got the following question: Show that the scalar product of two cartesian vectors $p_i\cdot q_i$ is invariant under coordinate transformations (orthogonal transformations) Now I know ...
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What is the way to represent a Lorentz tensor field?

For a vector field one can represent this with an array of arrows. There is a standard sort of way to represent tensors in Euclidean space as small ellipses. Is there any standard way of ...
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1answer
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Distinguishing between matrix forms when reordering indices of tensors

I'm studying general relativity and tensors. It seems that in the cooridnate independent form of the tensor, the order of indices matters even between an upper and lower index. For example, in general,...
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$)

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...
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2answers
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Seems like the coordinate independent 1-form transforms like a scalar

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
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1answer
56 views

Vectors transforming under change of coordinates

I was watching a lecture on tensors and the professor said that a defining feature of a vector $v$ is that it transforms under a coordinate transformation $x^{\mu} \rightarrow x^{\mu'}$ as $$v^{\mu'}...
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Extensors in mathematics and in physics [on hold]

Could someone explain in a simple but accurate manner what extensors are as mathematical entities and how they are used? How do extensors essentially differ from tensors? Are there or could there be ...
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How many moment of inertia about center of mass exist?

So imagine we have a rigid body and we want to find the moment of inertia about center of mass . Doesnt exist infinite axis that pass trough center of mass therefore infinte moment of inertia? Do they ...
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53 views

Levi-Civita tensor and the Lorentz group generators in the vector representation

In the vector representation of the Lorentz group its generators are given by - $$(J^{\mu\nu})_{\rho\sigma} = i(\delta^\mu_\rho\delta^\nu_\sigma-\delta^\mu_\sigma\delta^\nu_\rho)$$ It can be shown ...
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Relation considering a magnetic field irrotational

Considering a magnetic fiel $\vec{B}=B\,\vec{b}$ (along $\vec{b}$ direction), I try to find the followin relation : $$\vec{b}\times\nabla B/B = \vec{b}\times(\vec{b}\cdot\vec{\nabla})\vec{b}$$ and ...
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53 views

Divergenceless of energy momentum tensor for any metric $g_{\mu\nu}$

As suggested by @my2cts, from this post, I want to know if the divergenceless of energy-momentum energy tensor is valid for any metric $\eta_{\mu\nu}$ (i.e for example with $\eta_{\mu\nu}=g_{\mu\nu}$)?...
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3answers
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Doubt about the vacua equations of General Relativity

I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical. So, in a elementary point of view, ...
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A question regarding tensors [duplicate]

In Griffith electrodynamics book I ran into second rank tensors and when I searched it out on the web it gave variety of definitions and examples that I cannot understand. Can someone tell where is ...
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Riemann curvature in orthonormal frame and Lorentz transformations

I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, ...
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1answer
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Is the magnitude of the gradient of the tensor ellipsoid constant over the surface?

The following is from Lagrangian Dynamics by D.A. Wells: It can be shown that the direction cosines $l,m,n$ of a line drawn normal to the surface $\phi\left[x,y,z\right]=C$ are proportional to $\...
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1answer
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Why is principal strain parallel to principal stress for high symmetry crystals

Let say we obtain $\epsilon_{11} = S_{1111}\sigma_{11}$, for a single stress in a 11 direction, meaning that $E=\dfrac{1}{S_{1111}}$. Due to high isotropy of material the Poisson ratio is constant. ...
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1answer
40 views

How to lower both indices on the metric tensor?

If I have the tensor matrix $g^{\mu \nu}$ and I want the tensor matrix too $g_{\mu \nu}$, what is the calculation? Inverse? Adjoint? Some rule?
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1answer
61 views

Is there any identity for Levi-Civita contracting with 4 vectors?

Is there any identity to simplify $$\epsilon_{\mu \nu \rho \sigma}A^{\mu}B^{\nu}C^{\rho}D^{\sigma}$$ without explicitly putting into the indices?
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Divergence of rank 2 tensor in cylindrical coordinates

I am having trouble calculating the divergence of a rank 2 tensor $\sigma_{\mu \nu}$ in cylindrical coordinates using the Christoffel symbols. Given: $Div(\sigma)_\nu = \nabla_{\mu}\sigma_{\mu\nu} = ...
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3answers
86 views

Question about inner products of tensors and Einstein summation convention

So I am studying Special Relativity and basic tensor calculus and got stuck at an exercise. $$F^{\mu \nu}: = \left[ \begin {array}{cccc} 0&-{\it E_x}&-{\it E_y}&-{\it E_z} \\ {\it E_x}&...
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Priority of tensor product and inner product and outer product

In order to find the following calculation: σ x ( ( | ψ ⟩ ⊗ | ϕ ⟩ ) ( |...
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3answers
74 views

What is the real notion/feel of a tensor quantity? [duplicate]

I have been just introduced to the term tensor while studying Rotational Dynamics, particularly about Inertia. But I just don't get a clear line separating vector from a tensor. What does someone mean ...
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2answers
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Covariant and contravariant coordinates - index notation

I am learning about electrodynamics and have recently been introduced to the four vector. I also come fresh to the idea of covariant four vectors and contravariant four vectors. My question concerns ...
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1answer
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The action of Lorentz transformations on 4-vectors in special relativity

So I am studying special relativity and have been introduced to basic tensor calculus used in the theory. Recently, I came across a statement that is confusing me: $$\Lambda^\mu_{\,\,\nu} x^\nu = x^\...
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Proof: The sum of two tensors of order $n$ is a tensor of order $n$ [closed]

I can see that it's trivial for two vectors or matrices, when you're able to visualize it. It feels trivial for higher-order tensors too, but how should I start?
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Why is Penrose's diagrammatic notation for tensor operations not widely used? [closed]

Strictly speaking this is a mathematics question rather than a physics question, but since it is about a way of dealing with tensor bundles that is very remote from what is done in math, and very ...
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2answers
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Question about the true nature of the Spinor mathematical object [closed]

My question is kind of a silly one,but,I really would like to know what truly is a Spinor. I will explain what is my concept of "truly". Throught all the question post, consider finite vector spaces ...
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64 views

Most used convention about Christoffel symbols

Just a simple question: what is the most used form for Christoffel symbols, (1) or (2), see below: (1) $$\Gamma_{ij}^{k} = g^{kl}\Gamma_{lij}$$ and then, we have: $$\Gamma_{lij}=\Gamma_{lji}$$ (2) $...
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1answer
54 views

Show $2(B \cdot \nabla)B = \nabla |B|^2$ when the B-field is curl-less using summation notation

I was able to show for myself that $$ 2(\mathbf{B} \cdot \mathbf{\nabla})\mathbf{B} = \mathbf{\nabla} |\mathbf{B}|^2$$ when $\mathbf{\nabla} \times \mathbf{B} = 0$, but in order to do this, I had to ...
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Get relation from definition of stress-energy tensor and the conservation of energy

Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ...
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Tensors and the Klein-Gordon Equation

Consider the Klein-Gordon equation: \begin{equation} \frac{\partial^2 \psi}{\partial t^2} = c^2 \Delta \psi - \frac{m^2 c^4}{\hbar^2} \psi, \end{equation} and define for each one of its solutions $\...
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Symmetry of a tensor

This is from my notes, which I don't fully understand: It is straightforward to check that (anti)symmetry is a coordinate-independent notion, e.g., if the components of a tensor are symmetric in some ...
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2answers
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How do I look for (possibly) all coordinate transformations with a given metric?

From what I learned in tensor calculus so far, coordinate transformations are supposed to preserve the metric of the space. (Here I used GR notation, but the metric doesn't have to be the spacetime ...
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Silly question about kinematics and Christoffel symbols

An interresting "method" that allows you to know the acceleration vector with respect to any coordinate system is just a matter of recognize some key formulas. 1) Given the metric of a particlar line ...
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Doubt in notation of Variation with respect to a function

I cannot find this notation used anywhere on the internet or on SE (maybe I am searching using wrong tags). Hence, I am asking this question here. I don't know whether this even qualifies as a valid ...
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3answers
63 views

Meaning and significance of the Levi-Civita symbol

I am recently reading Sean Carroll's Spacetime and Geometry: An introduction to General relativity. I am much of a beginner but am really curios to learn about GR. In the first chapter, after ...
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Confusion about 4-velocity

I know that 4-velocity is defined as $$u^\mu = \frac{dx^\mu }{d\tau}$$ $$u_\mu=g{_\mu}{_\nu}u^\nu$$ and $$u_\mu u^\mu=c^2$$ Is it true that $$\frac{d\tau}{dx^\mu} = \frac{u_\mu}{c^2} = \frac{1}{c^...
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Covariant/contravariant manipulations for identity

On page 16 of Srednicki’s quantum field theory he says that we can get $$g^{\mu\nu}\Lambda^\rho_{~~\mu}\Lambda^\sigma_{~~\nu}=g^{\rho\sigma}$$ By starting with $$g_{\mu\nu}\Lambda^\mu_{~~\rho}\...
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4answers
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Uniaxial stress question

Let's have a rectangular profiled bar. Let us introduce force $\vec{F}$ which pull the bar apart. In the picture below let us make a virtual horizontal cut $A$. Well, everything is in the picture. ...
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I have been reading Mathematical methods for physicists (sixth edition) by Arfken and Weber and got stuck in section 2.9 Pseudo Tensors,Dual Tensors

In Mathematical methods for physicists (sixth edition) by Arfken and Weber the triple scalar product was defined as (in page 147-148): For $\vec A, \vec B, \vec C$ with components $A^i, B^j, C^k$ and ...
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1answer
54 views

Tensors in a two-dimensional Euclidean plane

Consider a two-dimensional Euclidean plane with coordinates $(x^1,x^2)$ If we define a set of new coordinate $z^1$ and $z^2$ $$z^1=x^1+ix^2$$ $$z^2=x^1-ix^2$$ A question is if a symmetric tensor $T^...
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Killing Spinor Equation in 4 Dimensions

From https://en.wikipedia.org/wiki/Killing_spinor we see that a Killing Spinor $\epsilon$ is defined as a solution of the following equation: $$\nabla_{X}\epsilon = \lambda X * \epsilon \quad , \quad ...
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Generalized divergence of tensor in GR

Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity: $$\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt{...
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Tensorial Proof that $\det(F^{\mu\nu})=(\vec{E}\cdot\vec{B})^2$?

I am trying to understand why $$\det(F^{\mu\nu})=(\vec{E}\cdot\vec{B})^2\tag{1}.$$ Of course one can just calculate the determinant of $F^{\mu\nu}$ expressed as a matrix with components given in ...
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2answers
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'Ambiguity' of dual vectors $\{dx^i\}$ in cotangent space in general relativity

The metric tensor is defined as: $$g = g_{ij}dx^i \otimes dx^j,$$ where I used the summation convention. We often omit the tensor product sign $\otimes$ and just write this as: $$g = g_{ij}dx^idx^...
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Relation between Levi-Civita tensor and the trace of Lorentz transformations

here is this tricky identity to prove in an appendix of W.B. Supersymmetry and Supergravity that's driving me crazy. Some premises first: This book use the van der Waerden's convention for spinor ...
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1answer
31 views

Variation of Antisymmetric tensor's trace?

Does an action of the form $$S=\int f(\text{tensors})g^{ij}\nabla_kA^k_{ij}\ d^4x$$ where $A$ is antisymmetric in the lower indices produce, upon variation, any nontrivial equation? Given that the ...
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Does entangled states must become non entangled states after the measurement?

In tensor notation. A state vector $|uv\rangle$ is a tensor product(non entangled states) if and only if there is $A\in E_1(u)$ and $B\in E_2(v)$ such that $A\otimes B$. So by postulate of quantum ...