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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Is potential energy a scalar operator?

If a scalar operator $\hat{S}$ is defined as an operator that is invariant under rotations, i.e $$U^\dagger S U = S,\,\,\,\,\,\,\, U=e^{-i\theta\hat{\mathbf{J}}\cdot{\mathbf{n}}}$$ which is ...
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Calculating the determinant of a metric tensor

Suppose the line element is $$ds^2 = -A(t,r)^2dt^2+B^2(t,r)dr^2+C^2(t,r)d\theta^2+C^2\sin^2\theta d\phi^2.$$ Since the metric is diagonal, to find the determinant I can multiply the diagonal entries, $...
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141 views

Tensor Analysis and SR [duplicate]

I'm looking for a textbook that covers, at least for a large part, special relativity with tensors/a geometrical approach. Most textbooks I have found develop tensors for the purposes of GR; I'd like ...
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The significance of the pressure term within the momentum-energy tensor [duplicate]

EDIT: this question is based around my notion regarding the possible role of potential energy in the momentum energy tensor T$_{\mu\nu}$, The answer below resolves the question and I have deleted ...
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213 views

Difference between $ \ F^{\mu\nu}$ and $\tilde F_{\rho\sigma}$

$ \ F^{\mu\nu}$ and the Hodge dual $\tilde F_{\rho\sigma}$ these are two tensors, related by $\epsilon_{\rho\sigma\mu\nu }$. My question is, is there any physical difference between them( I am aware ...
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Two Point Correlator

I have a problem to reproduce the following identity: \begin{equation} \Pi_{\mu\nu}(q^2) = i \int d^Dx e^{iqx} \langle 0 | T \{j_\mu(x) j_\nu(0) \} | 0 \rangle = (q_\mu q_\nu - g_{\mu\nu} q^2 ) \...
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Question about derivation of four-velocity vector

In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity. So, consider the following: For a massive particle ...
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828 views

Raising and Lowering Indices Question

Given $$\Gamma ^l_{ik}A^k_l$$ I want to lower/raise index $l$, I can insert $\delta ^m_m=g_{ml}g^{ml}$ Q1, is $\delta ^m_m=g_{ml}g^{ml}=4$? Q2, after inserting, it becomes: $$\Gamma ^l_{ik}A^k_l$$...
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How should the implicit sum $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted?

$C$ is a 3x3x3x3 tensor. How should the expression $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted? This is my guess: $$ \sum_{i=1}^3\sum_{j=1}^3 \sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}u_{i,j}u_{k,l} $$
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System of combined observables presented as tensor products

A state can be written as $$| \psi \rangle = \sum c_n | \psi_{nlm} \rangle$$ where $| \psi_{nlm \rangle}$ is the stationary states or eigenstates of the Hamiltonian in three dimensions (spherical ...
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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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Cubic symmetry and a stiffness tensor [duplicate]

Possible Duplicate: Stiffness tensor Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. How ...
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203 views

How are the *constant vectors* different from *vector fields* in terms of their respective transfomation properties?

How does one distinguish between the transformation properties of a scalar field $\phi(\textbf{r})$ or vector field $\textbf{A}(\textbf{r})$ (more generally, the tensor fields) from the transformation ...
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255 views

Proving a relation with Four-velocity tensor [duplicate]

I'm trying to show that: $U^a_{\space\space;b}U^bU_a = 0$ (Where U is four-velocity) and I'm stuck on how to go about it. I tried expanding it out into the Christoffel symbols, but that didn't seem ...
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Interaction between joint qubit quantum system [closed]

Consider the following interaction Hamiltonian $$H = \hbar \mu \sigma_{x} \otimes \sigma_x = \hbar \mu ( |01 \rangle \langle 1 0 | + |10\rangle\langle 01|)$$ acting on the joint states of qubits $\...
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Calculate Tensor and its transformations, transformation of derivatives of dependent variables

I had been learning tensor notation for a while and here's what's I have read: 1 Tensor had rank, denote two types covarient or contravariant. 2 $T(\__\alpha,\__\beta,\_\gamma)$ in place naming ...
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Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How does ...
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How can one prove $\int_V\text{d}V\ \vec A=0$, given that $\vec A\cdot \vec{\textbf{n}}=0$ and $\nabla\cdot\vec A=0$, without using tensor analysis?

In the course of learning electrodynamics, I was asked to solve the problem following: $\vec A$ is a vector which satisfies $\vec A\cdot \vec{\textbf{n}}=0$, where $\vec{\textbf{n}}$ is the normal ...