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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Simple four-vector partial derivatives

I am a beginner in tensor calculus, and am finding it difficult finding the result to what I assume are basic identities. I am trying to compute the following : $$ \partial_{\mu} x_{\nu} \quad and \...
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How to prove $\operatorname{div} \mathbf{A}=\operatorname{Div} \mathbf{A} \mathbf{F}^{-\mathrm{T}}$?

I recently focus on solid mechanics and I am reading Nonlinear Solid Mechanics A Continuum Approach for Engineering by Gerhard A. Holzapfel. However, I was confused by a mathematical formula eq(2.49), ...
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Lorentz Transformation of a Tensor

If I have the electromagnetic field tensor, then, under a Lorentz transformation: $$F^{'}_{\mu\nu} = \Lambda_{\mu}^{\alpha} \Lambda_{\nu}^{\beta} F_{\alpha\beta} $$ I know that the Lorentz matrix is ...
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Covariant formulation of electrodynamics homogenous Maxwell eq

It is know that $$\epsilon{^\mu} {^\nu} {^\rho} {^\sigma} \partial_{\nu} F_{\rho} {_\sigma} = 0$$ How can one deduce from this equation that $$ \partial_{\mu}F_{\nu} {_\lambda} + \partial_{\lambda}F_{\...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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How is tensor calculus applied to Einstein's field equations? [closed]

What is the relation between tensor calculus and Einstein's field equations? or What is the contribution of tensor calculus to Einstein’s field equations?
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Matrix “dimensional analysis” of Lagrangians in QFT

Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term. The electromagnetic Lagrangian (ignoring all constants ...
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Why can I use the Covariant Derivative in the Lie Derivative?

The Lie derivative is the change in the components of a tensor under an infinitesimal diffeomorphism. It seems that this definition does not depend on the metric: $$ \mathcal{L}_X T^{\mu_1...\mu_p}_{\...
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How do slanted indices work in special relativity? [duplicate]

What is the difference between $T^{\mu}{}_{\nu}$ and $T_{\nu}{}^{\mu}$ where $T$ is a tensor?
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Doubt on Tetrads, Energy-momentum tensors and Einstein's equations

Given, for instance, the perfect fluid energy-momentum tensor: $$T_{\mu\nu} = (\rho+p)u_{\mu}u_{\nu} - pg_{\mu\nu}\tag{1}$$ We can put (due to diagonalization procedure) into the diagonal for as: $$...
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Differential forms or Tensors for modern theoretical physics?

There many proponents to teaching differential forms and others teach with tensors. This is true for both mathematics and physics education. It seems mathematicians prefer to teach differential ...
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Physically measure the covariant and contravariant components of a vector?

I'm just wondering if there is a way to physically measure the covariant and contravariant components of a vector without prior knowledge of the metric. Suppose I have a speedometer of some sort to ...
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What is a coordinate-free formulation of deformation theory?

For example how are stress, strain and shear tensors described invariantly, without any coordinates, purely in a geometric manner? A formulation that avoids indices coordinates and matrices, even in ...
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Commutator of derivatives with torsion

I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. There in eq. 2.10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the ...
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Constructing the supertraceless portion of a connection over a supermanifold

Consider a tensor, $T$ of rank $(r,s)$ over a supermanifold, $M$ and take the supertrace over its indices $p$ and $q$ (DeWitt, p. 77, eq. 2.4.33): $$(-1)^{a_q(1+a_{p+1}+...+a_{q-1})}T^{a_1...a_{p-1}...
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How to calculate the number of independent components/degrees of freedom for symmetric tensors? [closed]

I was studying about the cosmological perturbation theory and came across this: ""Being symmetric, the two perturbed tensors contain ten degrees of freedom each, describing different aspects of ...
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Electromagnetic four-potential manipulation

I was looking at my special relativity notes and when covering EM, I was wondering if the following is true? Namely, given the four-potential $A_{\mu}$, by definition it is that: $F_{\mu \nu} = -A_{\...
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Inertia tensor in non-cartesian coordinates

given a rigid body $K$, I always had seen the formula \begin{equation} I_{ij} = \int_K[\mathbf{x}^2\delta_{ij} - x_ix_j]\rho(\mathbf{x})\mathrm{d}^3\mathbf{x} \end{equation} for the inertia tensor. ...
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Parameter Question In General Relativity

I am a math student taking a course in General Relativity. I haven't taken many physics/applied maths courses before, so I am not sure if I can describe this question well, but I am slightly confused ...
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Doubt on the precise definition of a general stationary rotating metric: the metric coefficients have which form?

Considering the following metric tensor $[1]$, with signature $(-,+,+,+)$, coordinates $(x^{0},x^{1},x^{2},x^{3})\equiv (t,r,\theta,\phi)$ and $c=1$: $$ds^{2} = g_{00}dt^{2} + g_{11}dr^{2} + g_{22}d\...
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Commutator of covariant derivatives acting on a vector density

Let $\mathfrak n^\alpha$ be a vector density of weight 1. Define the covariant derivative $\nabla$ such that under a coordinate transformation $x^\mu \to \bar x^\mu$ $$ \nabla_\rho \mathfrak n^\alpha ...
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Deriving the general formula of $\mathcal{\epsilon_{ijk}} \mathcal{\epsilon^{ijk}}$

As stated in the title, with $i,j,k=1,...,N$. I expanded $\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}$ as follows: $$\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}=\underbrace{\mathcal{\...
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How do I tensor differentiate a factor without tensors?

How do I tensor differentiate a factor without tensor, such as: $$\partial_\mu e^{i\Lambda(x)}\tag{1}$$ Should it be zero or should I differentiate it twice changing the order of the tensors follows:...
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Efficient method to evaluate the Christoffel symbols and Riemann tensor in Bondi-Sachs coordinates

In General Relativity we may employ the so-called Bondi-Sachs coordinates $(u,r,x^A)$ adapted to a null foliation. The level sets of $u$ are null hypersurfaces and $(r,x^A)$ are coordinates on the ...
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Tensor derivative in special relativity and fluid mechanics

I’m working through Special Relativity by V. Faraoni, and am puzzled by something in his chapters on tensors. He tells us that the partial derivative of a tensor field, e.g. $T_{\alpha, \gamma}$, is ...
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Doubt on proper time explicit integration

I have a doubt on explicit calculation of proper time. Considering that the metric is given by: $$ds^{2} = -Adt^{2} + B^{-1}dr^{2}+Cd\Omega^{2} -2Ddtd\phi \tag{1}$$ where $d\Omega^{2}$ is the solid ...
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How to express a rank-2 tensor as a spherical tensor?

A common example how to write a rank-2 tensor in the spherical basis is an outer product of two vectors, $$ T_{ij} = a_i b_j $$ such that $$ T_{ij} = \frac{\textbf{a}\cdot\textbf{b}}{3}\delta_{ij} + ...
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Tensor gymnastics

I am working on a "tensor gymnastics" exercise, and have arrived at the following line to simplify: $\delta_{ik} y^{i} X_{ij}$ where $\delta_{ik}$ is the Kroenecker delta. Does this simplify to: $y^...
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Recasting integrals from Lagrangian to Eulerian frame

Working on a research problem in the continuum mechanics of fluids. For clarity, uppercase will be used for tensors in the reference configuration, and corresponding spatial items will be in lowercase....
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Replacing the contravariant Minkowski metric tensor by its inverse

From Wikipedia, the Minkowski metric is defined (using (- + + +) signature) as : $$\eta_{\mu \nu} = \eta^{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ...
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Can we do better than “a spinor is something that transforms like a spinor”?

It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do ...
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Energy and spacetime: a doubt on notation

In the reference $[1]$ I saw a very neat formula, given by: $$ \mathcal{E} =: \int_{\Sigma} d^{3}x T_{00} = \frac{1}{8\pi G}\int_{\Sigma} d^{3}x G_{00}. \tag{1}$$ The author stated that this is the &...
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Evaluating $\sigma^{\mu\nu}F_{\mu\nu}=i\alpha \cdot E+\Sigma\cdot B$ matrix, spin dependent term in quadratic Dirac equation

I derive the quadratic form of Dirac equation as follows $$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\...
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Ricci tensor manipulation of weak field metric in General Relativity

I'm having some problems trying to work through my university's GR notes, regarding the derivation of the weak field metric from the perturbation of the Minkowski metric. I'll provide the relevant ...
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How to obtain the following pair correlation function

I came across this following pair correlation function. It says: the pair correlation function of velocity fluctuations, $\langle\delta u_\alpha(t;\pmb{r})\delta u_\beta(t';\pmb{r})\rangle$, is the ...
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Question about the definition for the scalar magnitude of a symmetric 2nd-rank tensor in a given direction

The scalar magnitude $S$ of a symmetric 2nd-rank tensor $S_{ij}$ in a given direction having direction cosines $l_i$ is given as: $$\tag{1} S=S_{ij} l_i l_j$$ This result is obtained by starting with ...
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Distinction between contraction and application of a tensor in abstract index notation

A somewhat similar question is this one but it is not quite the same. I am getting used to the abstract index notation used for tensor algebra. So far so good, but the is one issue that concerns me, ...
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A doubt on Christoffel symbols: to be a tensor, or not to be, what's the answer to that question? [duplicate]

Following $[1]$ we realize that, in order to construct a covariant derivative, we must to compare two possible covariant derivatives such as: $(\bar{\nabla}_{a} - \nabla_{a})\Omega_{b}$, where $\...
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Writing down a metric tensor given parametric equation of the surface

Let me begin by saying this question isn't related to GR. I'm reading a paper (see https://arxiv.org/abs/0903.0798v1) that talks about deriving a Schrodinger equation for an electron confined on a ...
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Compatibilty of spatial metric in Baumgarte's Numerical Relativity

There is an exercise 2.8 Baumgarte's Numerical Relativity (p. 32): Show that 3-dimensional covariant derivative is compatible with the spatial metric $\gamma_{ab}$, that is, show that $$ D_a \...
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How to calculate the field tensor from a metric?

Given a metric, for example $$ ds^2 = -A(r)dt^2 + B(r)dr^2 + C(r)d\theta^2 + D(r) d\phi^2, $$ and assuming that the fields go as $$ \textbf{E} = E(r)\hat{r} \quad \text{and} \quad \textbf{B}=0, $$ ...
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Is Ricci's theorem can be simply deduced using covariant derivatives of fundamental tensors? [duplicate]

Well Ricci's theorem is given by: $$\mathrm{D}g_{ij}=\mathrm{D}g^{ij}=0$$ I was wondering that if the theorem can be proved using covariant derivatives of $\delta_i^k$, $g_{ij}$ and $g^{ik}$. I ...
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Ordering of Tensor indices: a specific example

As an example of construcing tensors out of the product of other tensors I found the following example. $$T_{i}{}^j{}_{kl}=A_i{}^jB_{kl}+C_m{}^{mj}D_{ikl}$$ But I am a bit confused about the ...
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Matrix form of second order contravariant and covariant tensors

In Schaum Tensor Calculus Solved Problems 3.10 and 3.14 the following matrix equivalent of second order contravariant and covariant tensors are made, but without the derivation being explained ($\...
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Can a 1-form be the sum of two 1-forms?

I am interested in creating a Lie algebra-valued 1-form (gauge field/connection) for the Poincare algebra. Note that this algebra has two species of generators. In SU(2) Yang-Mills, the connection ...
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Doubt on a (“straightfoward”) derivation of Weyl tensor

I) My doubt: After Kruskal coordinates, we can introduce penrose diagrams after a quick talk about conformal metric tensors. Then, after the study of penrose diagrams the student should know that a ...
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Symmetries of double dual of Riemann curvature tensor [closed]

The definition of the double dual of Riemann is as follows: $$G^{\alpha\beta}{}{}_{\gamma\delta} = \frac{1}{2}\epsilon^{\alpha\beta\mu\nu}R^{\rho\sigma}{}{}_{\mu\nu}\frac{1}{2}\epsilon_{\rho\sigma\...
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Using tensor calculus in thermodynamics : [closed]

This is my first post here in Physics Exchange, I hope I'll find my questions answers here. As you all read in the title "Using tensors in thermodynamics" basically that was a problem in the ...
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Another question of index notation in tensor calculus

I have this equation $$\nabla_{a}(g_{bc}\lambda^{c})=(\nabla_{a}g_{bc})\lambda^{c}+g_{bc}\nabla_{a}\lambda^{c}$$ And making some calculations $$ \lambda^{c} (\nabla_{a}g_{bc})= \nabla_{a}(g_{...

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