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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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How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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Am I proving an identity about Maxwell's “Magnetic” Equations correctly?

Question 3.12(d) of Gravitation (MTW) has me show Maxwell's "magnetic" equations $F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta} = 0$ can be rewritten as $F_{[\alpha ...
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Mathematical software for calculating Christoffel symbols, curvature, etc [duplicate]

I'm looking for a Mathematical software that can do the following: I put in a metric (some kind of a two-dimensional matrix), and the software calculates Christoffel symbols, Ricci curvatures, ...
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Collision between two identical particles

I was working on the exercises of the identical particles chapter of Cohen-Tannoudji, and got stuck due to some conceptual flaws. My questions are numbered below. In the problem, there are two ...
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Write electromagnetic field tensor in terms of four-vector potential

How can we know that the electromagnetic tensor $F_{\mu\nu}$ can be written in terms of a four-vector potential $A_{\mu}$ as $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$? In the ...
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Dual space and Metric tensor

So I know that the dual space is the set of all linear transformations that map a vector from a vector space to the field of the space itself (the real number line, complex, quaternions). From YouTube ...
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Do Bianchi identities hold in all coordinates?

I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
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Tensor index in special relativity?

I'm studying special relativity and I have some difficulties with tensor index. Take for example the Lorentz matrix, whose elements are written as $\Lambda^\mu{}_\nu$. $\Lambda^\mu{}_\nu(v) = \...
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Can thermodynamical work be a function of state?

following the question: Why dW=pdV is an inexact differential? Usually the pressure p is given by the equation: $$p=-\left.\frac{\partial U}{\partial V}\right|_{S},$$ where $U=U(S,V)$ is internal ...
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Navier-Stokes : divergence or covariant derivative of a tensor : 1 vector result?

I don't understand very well the following definition concerning Navier-Stokes equation : where $\vec{u}\otimes\vec{v}$ is a tensor (2,0), isn't it ? This is not scalar since $\vec{u}\,\vec{v}^{T}...
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Proof that the scalar curvature of a two-dimensional space can be expressed by only one component of the Riemann tensor

So I'm working on a question that asks for a proof that in two-dimensional space, the scalar curvature is given by: $$R = \frac{2R^1{}_{212}}{{g}_{22}}$$ Now, I've been playing around with the ...
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Equality between derivatives of the metric

In one of my lecture, it is said: Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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Why isn't the scalar product of a covector and vector symmetric? [migrated]

In tensor math, how come the scalar product of a co-vector (co-variant vector) with contra-variant vector, as written between angle bracket separated by comma, $\langle x, a \rangle$, is not symmetric?...
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Product of $N$ Pauli matrices

I am trying to perform diagonalization on an Ising Hamiltonian, with $N$ spins, containing a term of the form \begin{equation} \sum_{i = 1}^{N} \sigma_i^{x} = \sum_ {i=1}^{N} 1_1 \otimes \cdots 1_{i-1}...
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Derivative of tensor product of quantum states

Recently I asked a question over at the math stack exchange: https://math.stackexchange.com/q/3210375/. However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
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Questions about special relativity, index in the Lorentz matrix

I'm studying special relativity I have read this: We have $ x^u = (ct, x^1,x^2,x^3) $. If we apply Lorentz transformation we can write: $x'^u = \Lambda^{u}_{\hspace{0,2 cm}\nu} x^{\nu} $ $x'_u =...
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Quantum mechanics angular momentum spherical tensor components

In Sakurai Quantum Mechanics, problem 3.25b we imagine $J_z^2$ as the component of a tensor with components $T_{ij} = J_iJ_j$. $J_z^2 = \frac{1}{3}\pmb{J}^2 + (J_z^2 - \frac{1}{3}\pmb{J}^2) $ The ...
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How do I derive the relationship between the classic magnetic field and the electromagnetic tensor?

I am trying to derive the relationship between the components of the electromagnetic field tensor and the components of the magnetic field by inverting $F_{ij} = \epsilon_{ijk}\beta_{k}$. The ...
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Doing addition and subtraction with tensor diagrams?

Tensor diagrams are a beautiful and useful tool for making calculations with tensors, up until you need to contract with the sum or difference of two tensors, at which point it seems to become awful. ...
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Tensor product of photon number states

I'm looking to compute the tensor product of photon number states. I suspect this is a fairly simple quantum optics problem, but am having the following problem. Consider a qubit which is in the ...
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Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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Christoffel's Symbol's relation to the Metric Tensor

In chapter 9.2 of "Tensors, Relativity and Cosmology", the contracted Christoffel symbol of the second kind as a function of the metric tensor was defined as: $$\Gamma_{nm}^m=\frac{1}{2}\left(g^{mk}\...
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Is polarization matrix always diagonalizable?

In chapter 31 of Feynman lectures Vol 2, he covers polarization , polarization tensor and its diagonalisation, he proves that for a crystal, the tensor matrix is symmetric hermitian and hence ...
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Covariant surface vector

On pg 74 of Dalarsson's Tensors, Relativity and Cosmology (The Integral theorems for tensor field chapter), the covariant surface vector was defined as: $$dS_k=\frac{1}{2}\epsilon_{kmn}dx^mdx^n=\frac{...
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Curl of Vectors

On pg. 71 of Dalarsson’s Tensors, Relativity and Cosmology 10.4 Curl of Vectors In an arbitrary $N$-dimenstional metric space the curl of a vector function $A_m$ is a second-order covariant ...
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Earth's surface area

Here we are trying to calculate the earth's surface area via geodetic coordinates: \begin{align} x&=(R\,p(\lambda)+h)\sin\lambda\,\cos\phi\\ y&=(R\,p(\lambda)+h)\sin\lambda\,\sin\phi\\ z&...
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Indices in Special Relativity

I am studying special relativity and I can't figure out what is the difference between the matrix - index notation between: $$ Λ_{α}{}^{β}, Λ^{α}{}_{β}, Λ^{αβ} ,Λ_{αβ} $$ Why do we introduce this ...
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Rotation matrix - levi-civita symbol

I'm trying to solve the following problem: Given a rotation matrix $R_{ij}$, show that $$n_k=\frac{-R_{ij}\epsilon_{ijk}}{\sqrt{(3-tr(R))(1+tr(R))}}$$ and that $$\sin(\phi)=-\frac{\epsilon_{ijk}...
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Riemann tensor Contracted with full antisymmetric tensor

I'm not able to show that $\epsilon^{abcd} R_{bcae} = 0$ Note: Properly, I have to show that $\epsilon^{Iabc} R_{abIL} = 0$, where $I,L$ are tetrad index and $a,b,c$ are spacetime index, but it ...
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The chain rule and velocity transformation in relativity (2)

First of all, these answers (How to derive the law of velocity transformation using chain rule?, The chain rule and velocity transformation in relativity, and other from a quick search on this site.) ...
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Is the addition of a Christoffel symbol and the partial derivative of a vector a tensor?

The partial derivative of a vector $V^\lambda , _\nu$ is not a tensor. Neither is a Christoffel symbol $\Gamma^\lambda _{\mu \nu}$. Is the addition of these two objects a tensor? If they were ...
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Linearity of Maxwell's equations in tensor formulation

Maxwell equation in tensor formulation are $\partial_\nu F^{\mu \nu}=J^\mu $ and $\partial_{[\gamma} F_{\mu \nu]}=0$. So to show Maxwell equation are linear in vacuum is the following method correct: $...
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Different between index locations on tensors

My question is in regard to the position of upper and lower indices on tensors, specifically in this case I am considering position 4-vectors and the Minkowski matrix. On the Wikipedia page I see ...
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Is the metric $ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu$ Lorentz invariant?

Postulate of Special Relativity leads to the conclusion that the metric in flat Minkowski space $$ds^2=c^2t^2-\textbf{r}^2=\eta_{\mu\nu}dx^\mu dx^\nu\tag{1}$$ is Lorentz invariant. This follows as a ...
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0-rank tensor vs vector in 1D

What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D? As far as I understand tensor is anything which can be measured and different measures can be transformed into ...
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Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?

I've come across several references to the idea that to upgrade a law of physics to general relativity all you have to do is replace any partial derivatives with covariant derivatives. I understand ...
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Riemann curvature tensor components having 3 or 4 distinct components

When, if ever, will we see Riemann curvature tensor (RCT) components having 3 or 4 distinct indices?, like for $R_{txxy}$ or $R_{txyz}$ for ex. How this came about was I that I was reading that ...
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Proving co-rotated time derivative is objective

So I need to show that: $$ \mathring{u}^+ = Q\mathring{u} $$ My progress is thus far, since a vector is objective (i.e. $u^+=Qu$): $$ \mathring{u}^+= (\mathring{Qu}) = \dot{(Qu)}-w(Qu)=\dot{Q}u+Q\...
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Can we divide tensor components?

The direct product of tensors can be used to create new, higher-rank tensors. As in: \begin{equation} A^i_{\text{ }j}B^{kl}=C^{i}_{\text{ }j} \text{ }^{kl} \end{equation} If two tensors have $A_{\mu ...
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How does the partial derivative of a tensor of rank $n$ creates a tensor of rank $n+1$? (cartesian coordinates)

The partial derivative of a tensor of rank $n$, $T_{...i}$, with respect to $x^j$ can be expressed using the transformation rule: \begin{equation} \frac{\partial}{\partial x^j}T'_{...i}=\frac{\...
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Doubts about the use of tensor product In quantum mechanics

I'm studying quantum mechanic in particular tensor product and Hilbert space (for the first time). I have some doubts and I would like to check if I have understood correctly. Factorization The ...
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What does $\nabla_{[a}\Omega_{b]}=\nabla_{[a}\nabla_{b]} t=0$ represent?

In Sec. 2.3 of Baumgarte and Shapiro's "Numerical Relativity", we find this statement: From $t$ we can define the 1-form $$\Omega_a = \nabla_a t, \tag{2.19}$$ which is closed by construction, $$\...
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Is the antisymmetrisation of $a^{\mu}b^{\nu}\epsilon_{\mu\nu}= a^{[\mu}b^{\nu]}\epsilon_{\mu\nu}$ with antisymmetric tensor $\epsilon$ mandatory?

When in tensor algebra the product of 2 vectors with a antisymmetric tensor appear, is antisymmetrisation compulsory ? Given an antisymmetric tensor $\epsilon_{\mu\nu}$, is $$a^{\mu}b^{\nu}\epsilon_{...
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Anti-symmetrization brackets break Einstein summation convention

How does one properly evaluate something of the form $$ g_{a}^{\, [b} R_{c] b}~? $$ when I try to expand using the definition of anti-symmetrization brackets the Einstein summation seems to break: $$...
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Inverse of metric tensor

The Minkowski metric tensor have the relation $\eta_{ij} \eta^{jk}=\delta_i {^k}$. That is the inverse of the Minkowski matrix is the matrix itself. Analogously, is it true that $g_{ij} ...
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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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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 ...