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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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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Rotation calculation of an imbalanced object

Let's assume I have a scale like object hanging from a point. Now if I put an object inside it stays as it is, as we have replaced the spring with some sort of fabric, that doesn't expand itself. ...
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Relation bewteen to different operators in different charts

Let $\phi$ be a coordinate system and $\partial/\partial^{\mu}$ and $dx^{\mu}$ be the associated coordiantes bases. Then in the region covered by these coordinates we can define a derivative ...
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Contravariant components $a^k$ of the acceleration $\boldsymbol{a}$ of a particle

Well I was studying the uses of Christoffel in physics and I got the following Idea : Let us consider a particle moving on a trajectory defined by spherical coordinates $r, \theta, \varphi$, the ...
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Is the off-diagonal part of this rank-2 tensor integrand odd?

Peskin and Schroeder in Introduction to Quantum Field Theory consider the following tensor integral (Eq. 6.46): $$\int \frac{\mathrm{d}^4l}{(2\pi)^4} \frac{l^\mu l^\nu}{D^n} = \int \frac{\mathrm{d}^4 ...
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Area of parallelogram in spacetime calculation and its dependence on reference frame

I encountered the following equation in "Gravitation and Cosmology" by S. Weinberg in the section of Riemann Curvature Tensor. $\oint x^\rho dx^\nu=\delta a^\rho\delta b^\nu-\delta a^\nu \delta b^\...
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Does $\partial^\mu X^\nu = \partial_\mu X_\nu$ (Neumann Boundary Conditions)?

Problem I am trying to prove that the Neumann boundary condition , $\partial_\sigma X_\mu=0$ , implies that no momentum flows out of the end of an open string. I'm told that the associated conserved ...
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How Does Four normal projection of 4D Riemann tensor vanish?

I am studying numerical relativity from the books of Baumgarte & Shapiro. At the page 39 (eq. 2.86), it is said that $$n^{p}n^{q}n^{r}n^{s}~{}^{(4)}R_{pqrs}$$ vanishes, where $n^a$ is normal ...
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What is the real Physical Meaning of tensor?

I have read something about tensor calculus from Arfken and Corvet but all of them are more some mathematical algebra for tensors. what happens in reality and nature when we use tensors in our ...
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Is it possible to write the Hilbert Action as a product of two identical tensors?

We know the Maxwell action can be written as the tensor product of the tensor $F^{ab}$ with itself. $F^{ab}F_{ab}$ [Edit: This bit I forgot to mention in the original quesiton] Using the product rule ...
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Inverse of a metric tensor on a Hermitian manifold

Let $(M, g)$ be a Hermitian manifold. We have a metric tensor $g^{i \bar j} dz_i \otimes d\bar{z_j}$, where $(g_{i \bar j})$ is a hermitian positive definite matrix. Now we naturally get the inverse ...
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Difference between scalars, vectors, matrices and tensors [duplicate]

What is the difference between a scalar, vector, matrix and tensor in simple terms? Are vector fields and tensors related?
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Covariant derivative of a metric determinant

The covariant derivative of a metric is zero $g_{\alpha\beta;\sigma}=0$. Is the covariant derivative of a metric determinant zero following the assumption($g_{\alpha\beta;\sigma}=0$): $$ g_{;\sigma}=...
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Confusion regarding the terms in the covariant derivative of a Tensor

I am learning General Relativity from Leonard Susskind's Lectures. In Lecture three, he introduces to covariant derivatives, and I understood it's meaning. But when he applies it to a Tensor, I am ...
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On the usage of the permutation tensor

The permutation symbol $\epsilon_{ijk}$ is a set of 27 numbers, of values +1, -1 or 0 depending on $i, j, k$. I am familiar with the usage of $\epsilon_{ijk}$ in the context of vector products, such ...
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The form of QED Vertex Correction

In chapter 6 of Peskin-Schroeder's text Introduction to Quantum Field Theory it is argued that the form of vertex correction for QED can only have the following form (since we have only the constants, ...
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What's the action of a tensor field on a scalar field?

We say that a tangent vector, eats a smooth function and produces a number. This is also intuitively clear. What does a tensor spit out when it eats a smooth function? I know that a (k,l)-tensor eats ...
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Question solving tensor problems for the Special Conformal Killing Equation

Background I know that following index notation, these are true: $$\partial_\mu x^\nu = \delta^\mu _\nu \hspace{5mm} and \hspace{5mm} \partial_\mu x_\nu = \eta_{\mu\nu} \tag{1}$$ Exercise Knowing ...
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Does the order in which the Kronecker delta is in a tensor product situation matter?

I know that the Kronecker delta is used to raise and lower indices, but I am not certain if the order in which it is placed in an equation matters, what I mean is, for example: Are the following two ...
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Irrep of stress energy tensor

We have 4-tensor of second rank. For example energy-momentum tensor $T_{\mu\nu}$, which is symmetric and traceless. Then $T_{\mu\nu}=x_{\mu}x_{\nu}+x_{\nu}x_{\mu}$ where $x_{\mu}$ is 4-vector. Every ...
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Notation of Mixed Tensors: Risk of Confusing Index Positions?

The convention for notating indices of a tensor is to write a contravariant index superscript and a covariant index subscript. If one has a pure contravariant or a pure covariant tensor of $2$nd order,...
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Action of a superconnection on a vector field

On page 4 of $\textit{Natural and Projectively Invariant Quantizations on Supermanifolds}$ by Leuther & Radoux (https://arxiv.org/abs/1010.0516v2), they write down the action of a superconnection ...
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Covariant Differentiation on a Riemann Tensor

Working on an assignment I came across the following problem: Show explicitly the Bianchi identity: $$ R^{a}_{b[cd;e]} = 0 $$ where ; denotes covariant differentiation $$ R^{a}_{bcd} = \frac{1}{2}g^{...
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Covariant vs contravariant vectors

I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. It seems to me that any vector can be decomposed in either of those basis, thus one can have ...
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Second-order tensors as linear operators

A tensor is formally defined as an object whose components obey some transformation rules. I, however, find it more intuitive to look at (second-order) tensors as a linear operator/function between ...
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What is the best way to imagine the difference between vectors and one-forms?

I am studying the GR and reading the Schutz. He is defining the one-form as $\widetilde{p} = p_{\alpha}\widetilde{w}^{\alpha}$, and a vector $\vec{A} = A^{\beta}\vec{e}_{\beta}$ such that $$\...
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Alternative formulation for the gradient of divergence of a vector

I am trying to derive two identities which are needed in simplifying both solid and fluid momentum balance equations. Let $A$ be a second-order matrix and $u$ be a vector with three Cartesian ...
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What is the operator $I_\zeta$, satisfying $ d I_\zeta + I_\zeta d = 1$ that allows us to define Noether-Wald charge?

In these lectures on general relativity by Geoffrey Compère, the author, in section 1.3.3., equation 1.62, define an operator $I_\zeta$ which satisfies the identity $$ d I_\zeta + I_\zeta d = 1.\tag{...
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Calculating the metric tensor

In my lecture we just approached the metric tensor and the general form of a scalar product. So for two vectors $\vec{x}$ and $\vec{y}$ the scalar product is $\vec{x} \cdot \vec{y} \enspace = \...
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Lorentz Invariance of the Euler-Lagrange equation for fields

Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
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What are the rules when differentiating tensor functions to the power of 2?

How do I differentiate tensor squared functions? I know that, for example, to differentiate a function like $x^\rho x^\mu$ it is as follows: $$\partial_\mu (x^\rho x^\mu) = \frac{\partial(x^\rho x^\...
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Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? [duplicate]

Is the result of the product of metric tensors $\eta^{\mu\nu} \eta_{\mu\nu} = 1$? If so how would I prove this? I know that tensors are represented as matrices but I don't know how I'd prove this (...
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Effect of Levi-Civita symbols on rank-two tensors

I am trying to understand the following: $$ \epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{...
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Is there any meaning to the object $\partial_{\mu} \Gamma^{\rho}_{\ \nu\sigma} + \Gamma^{\rho}_{\ \mu\lambda} \Gamma^{\lambda}_{\ \nu\sigma}$?

In a calculation I am encountering the object $$ O^{\rho}_{\ \mu\nu\sigma} \ := \ \partial_{\mu} \Gamma^{\rho}_{\ \nu\sigma} + \Gamma^{\rho}_{\ \mu\lambda} \Gamma^{\lambda}_{\ \nu\sigma} \ , \tag{1}$...
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If $S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma}$, then is it true $S_{\mu\nu\sigma} = T_{\mu\nu\sigma}$?

For any vector $V$, suppose that the following equality holds $$ S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} $$ for two tensors $S$ and $T$. Does it follow ...
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Every symmetric second rank covariant tensor can be transformed into diagonal form with diagonal elements 0,±1

I started learning about tensors, and this theorem was mentioned: "Every symmetric second rank covariant tensor can be transformed into diagonal form in which the diagonal elements are either 1,0 or −...
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I want to know why engineering strain is not a tensor

One of the most classical examples in the mechanics of materials is that engineering strain is not tensor. I want to know why the engineering strain doesn't meet the tensor definitation. ...
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63 views

About variation of Ricci tensor

I have been doing some calculation on variation of Ricci's tensor with respect to the metric, that, according with S. Carroll (An Introduction to General Relativity: Spacetime and Geometry, equation 4....
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How can I show the Einstein Tensor using second Identity of Bianchi? [closed]

The Einstein tensor given by: $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$ Can be shown using Bianchi identity?
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How can I prove that $\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$? [closed]

I want a simple proof of this identity: $$\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$$ If there's no answer, give me a hint or something would help to prove it, and thanks!

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