Questions tagged [tensors]

For questions about tensor, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.

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Derivative of exponential tensor with respect to a vector

Suppose you have a rank 4 Tensor $T$ whose coordinates are, in some basis, $T_{ijkl}$. Say the indices take values in some representation of $T$. Now suppose I construct a matrix by dotting this ...
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How to evaluate these tensor operations?

I'm struggling to understand how to evaluate these expressions, despite being given their definitions inside the question. Can someone please help? EDIT: is the first one correct. I have just flipped ...
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Confusion with tensor algebra in the context of special relativity and electrodynamics

I'm so confused with tensor algebra and I need help in finding the answer to part b. I honestly cannot figure out how to multiply tensors. Can someone please help/explain?
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Induced mapping between tensor algebras

I'm dealing with tensor algebra in my differential geometry classes, and in particular with induced mappings. The mapping defined is the following: If $f:V\longrightarrow W$ is an isomorphism, then \...
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Prove that $\bf RAR^T = A$ where $\bf A$ is a symmetric 2nd order tensor and ${\bf R} = {\bf I} - 2{\bf e} \otimes {\bf e}$

Here's how I've gotten so far $${\bf RAR^T} = {\bf A} - 2{\bf A} ({\bf e} \otimes {\bf e}) - 2 ({\bf e} \otimes {\bf e}) {\bf A} + 4({\bf Ae}\cdot{\bf e})({\bf e} \otimes {\bf e})$$ I can't quite ...
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Tensor identity in orthogonal coordinate systems

In Riley, Hobson & Bence's "Mathematical methods for physics and engineering" third edition, in the chapter about tensors, one of the exercises involves finding the expression for the ...
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Why do we need to use the chain rule in the transformation of a partial derivative of a vector component from different coordinate systems?

Let us consider the partial derivative of a vector component $A^i$, this is $\partial_jA^i = \frac{\partial A^i}{\partial X^j}$ We also know that we can write the transformed contravariant vector ...
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What does a matrix indexed with two pairs of indices mean?

I came across the following in a paper: "... we form the matrix $\mathbb{B}$ with entries $$ B((\hat{j}, \hat{l}), (j, l)) = \sum_{l_1}^{R_A} \sum_{l_2}^{R_A} \left( \sum_{j'}^M A_k^{l_1} (j', \...
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Tensor Calculus Problem

With the following convention for the antisymmetric basis of forms: $\boldsymbol{\tilde\omega}^{[\mu_1}\otimes ...\otimes\boldsymbol{\tilde\omega}^{\mu_p]} = \frac{1}{p!} \Sigma^{p!}_{i=1}(-1)^{\pi(i)}...
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Show $(1,1)$ tensor acts as a map from vectors to vectors

In Sean Carroll's Spacetime and Geometry book, pg. 22, it was stated that a (1,1) tensor acts as a map from vectors to vectors: $${T^\mu}_\nu: V^\nu\rightarrow {T^\mu}_\nu V^\nu.$$ I tried showing ...
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Is it possible to reverse engineer the metric to find the basis vectors of the manifold?

For a given surface that has metric components $g_{\mu\nu}$ is it possible to find it's basis vectors and more importantly the exact surface that space is on. Since we know that $$g_{\mu\nu} = e_\mu \...
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What's is the best textbook to study tensor calculus for physicists?

I've finished studying tensor analysis from "Mathematical Methods in the physical sciences by: Mary L. Boas", but I wanna go furthermore in tensors that to be ready for General Relativity, ...
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Can the sum of this tensor's elements be simplified down to a concise expression?

A brief bit of background: The density $\rho$ of dislocations within a crystal as a result of shearing due to an applied strain is given by the sum of the elements of the Nye tensor $\alpha$: $$ \rho=\...
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How to prove that a Riemann Manifold is conformally flat?

In Ray D'Inverno's book - "Einstein's Relativity" he states a theorem that says the following - "Any two-dimensional Riemann manifold is conformally flat". How can I prove this? I ...
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$ \sum_{i_{R+1},…,i_K=1}^N δ^{i_1,…,i_R,i_{R+1},…,i_K}_{j_1,…,j_R,i_{R+1},…,i_K}=\frac{(N-R)!}{(N-K)!}δ^{i_1,…,i_R}_{j_1,…,j_R} $

What shown below is a reference from Introduction to vectors and tensors by Ray M. Bowen and C.C Wang. Precisely it is de definition of the generalised Kroneker delta thus if you know it you can even ...
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Can tensors be used to describe things that are not related to space or time?

Can tensors be used to describe things that are not related to space or time? In my understanding a tensor is a set of x, y, z, etc. or axis where each axis represents an array of object say [Paul, ...
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Quadruple Scalar Product - Proof Verification with Levi-Civita Notatation

To Prove: $$(\bar{a} \times \bar{b})\cdot (\bar{c} \times \bar{d}) = (a\cdot c)(b\cdot d)-(a\cdot d)(b \cdot c)$$ Here is my proof: $$(\bar{a} \times \bar{b})\cdot (\bar{c} \times \bar{d}) = \...
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How to find the metric tensor for a intrinsic transformation

Since it is possible to map a flat plane into 3d space through: $$X=G_X(u,v)$$ $$Y=G_Y(u,v)$$ $$Z=G_Z(u,v)$$ Then you can find the tangental basis vectors through: $$\frac{\partial}{\partial u}=\frac{\...
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Dilemma with tensors

I was reading lots of books lately on field theory, and unfortunately, a particular step in a calculation is causing me trouble. I cannot find the details of it in any book, and I couldn't derive it. ...
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Relating co-variant basis of different coordinate systems

In page 77 of Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfield, He relates a covariant basis in one coordinate with another. He does it in the following way: Say ...
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Riemann curvature tensor identity proof

Let $R_{abcd}$ be the Riemann curvature tensor defined by a torsion free and metric compatible covariant derivative. Ultimately I am trying to prove that $R_{abcd} = - R_{abdc}$ The prove in my ...
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How do cross products and bivectors transform?

Let $V$ a vector space over $\mathbb{R}$. As far as I understand, a bivector is an antisymmetric second rank tensor, thus an element of $V\otimes V$. Let $\{\underline{e_i}\}_{i=1}^n$ and $\{\...
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Show that the trace $c_{ii}$ of a quadratic form Q does not depend on the choice of basis (invariant)

So, I want to Show that the trace $c_{ii}$ of a quadratic form Q does not depend on the choice of basis (that is trace is invariant). I have the following information and ideas : I would like to try ...
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Differentiate scalar expression with a few nested matrix functions with respect to a matrix

I am trying to differentiate a scalar expression $u$ with respect to a matrix $X$, which has applications to Material Strength models. I've gotten through most of the work here, but am stuck at the ...
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Levi-Civta symbol question

$$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kl} (\delta_{jl}\delta_{ki} - \delta_{ji}\delta_{kl})$$ $$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kj}\delta_{ki} -\delta_{ii}\delta_{ji} = ...
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How to diagonalize a 2nd rank symmetric covariant tensor?

I've been self-teaching myself tensor mathematics by reading PDFs online, and I came across one that states that every 2nd rank symmetric covariant tensor can be diagonalized into a form where the ...
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30 views

Computation of Laplace operator of Einstein tensor

I'm new to Relativity and I'm trying to understand this computation: $$ (\nabla^\mu G)_{\mu\nu}=0 $$ where $G=Ric-\frac{1}{2}\mathcal{R}g $ is the Einstein tensor, and $\nabla^\mu=\nabla_{\partial^\...
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Product Rule for 2nd Order Tensors

Lets take three second order tensors $ A $, $ B $ and $ C $ . What is the derivative of $ABC$ w.r.t to a vector $x$. To me it seems like the general derivative rule does not follow here. Edit 1: $ A $ ...
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What is the name of this multiplicative tensor operation that can retrieve components?

I know this might be somewhat of a dumb question, but I've discovered a neat way to represent the components of a (assume real valued) tensor via a sort of "multiplication". Let's say I have ...
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Double Derivative of a vector valued vector function

I have an expression where $ g(A) $ is a vector valued vector function and A & X are vectors. $(d/dX)*(dg(A)/dA)$ I had to calculate the above expression. So what I did was, I changed the order of ...
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Transpose of a 3rd Order Tensor and its Symmetricity

Let B be a second order tensor and x be a vector. Then L is a third order tensor such that $ B=Lx $ and L is $ L=L_{ijk} e_i ⊗e_j ⊗e_k $ Find $ L^T $ and whether L is symmetric or not ? Any directions ...
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Where can I find a good course on tensor/Ricci calculus not focused on applications and physics?

Where can I find a good course on tensor/ricci calculus not focused on applications and physics? I've been running into lots of tensor-theoretic stuff in differential geometry, so I don't know if ...
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Tensor Calculus swapping out repeated indices question

Consider the following result from General relativity: $$(1) \space\space\Gamma^{j}_{ij} = \frac{1}{2} g^{jk} (\partial_i g_{jk} + \partial_j g_{ki} - \partial_{k}g_{ij} ) $$ My question is whether it ...
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Differentiating a summation with respect to a particular index

Consider the following Lagrangian: $$L = \sqrt{g_{ij}\dot{X^i} \dot{X^j}}$$ Note that in the above Lagrangian we have taken the Einstein Summation convention and the metric $g_{ij}$ is a function of ...
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Tensor identity proof

I have the following identity to prove: $$\operatorname{grad} \operatorname{div}\textbf{A}=\operatorname{rot}\operatorname{rot}\textbf{A}+\operatorname{div}\operatorname{grad}\mathbf{A}$$ The part ...
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Variation of metric

I am looking at the derivation of the Einstein field equations as the Euler-Lagrange equations of the Hilbert functional. To do this one starts with a variation $$ g(t) = g+th $$ of the metric, where $...
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Tensor product in Paul r halmos book

This is the definition of tensor product in the book. But my question is "Does for every $z\in \scr U \otimes \scr V$ there exists $x\in \scr U $ , $y\in \scr V$ such that $z = x\otimes y ?$"...
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Derivative of a matrix with a vector

Let there be three matrix F ,A (size (t+1) * t) and B (size t * t) and B is a function of x and x (size n) be a vector. $ F=AB(x) $ Find the derivative of F w.r.t x. When I tried to find the ...
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Ricci Curvature and Sectional Curvature

If we define the Ricci curvature for an orthonormal frame, we can simplify the sectional curvature formula to get: $$Ric(v, v) = \sum R(e_i, v)v \cdot e_i$$ However, I obtain a different formula to ...
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How to understand the indexing in Tensor product?

Recently, I learn the tensor product in physics. Here are some equations I meet: $$X_{\gamma \omega \rho \sigma}=\sum_{\alpha, \beta, \delta, \nu, \mu} A_{\alpha \beta \delta \sigma} B_{\beta \gamma \...
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elastic stress tensor under a given Riemannian metric

I am curious about the expression of the elastic tensor under a metric tensor, rather than the standard Euclidean metric. I have been reading an article,which gives the expression about the stress ...
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56 views

Notation for a particular tensor field

Suppose that $E$ is a bundle with metric $\langle\cdot,\cdot\rangle$ over a manifold $M$ and consider an $E$-valued $1$-form $A \in \Gamma(T^*M \otimes E)$. We can write $A$ in coordinates as $$ A = ...
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Difference between Identity Matrix and Identity Tensor

I was thinking whether the identity matrix could be considered as a rank 2 tensor or not, but then I found out that there is a tensor called identity tensor that has the same function. So what do you ...
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Linear Regression : functional multivariate or tensor

What would be the difference in treating data as tensor or as multivariate functional data? I have, for N days, the Energy Price Offer Curve for each hour of the day; I could see these data as N ...
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49 views

A weird exercise for proving a tensor

I have the following exercise: Give a variant arbitrary values in some coordinate system and let it transform by the proper tensor rules. Show that this construction produces a tensor. More ...
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28 views

Basis vectors and one-forms in curvilinear coordinates - normalized or not?

Here's my dilemma. We know, that $\partial_i$ forms a basis of vector fields and $\mathrm{d} x^i$ forms a basis of co-vector fields, so that we can write, in general (vector and covector) $$ v = v^i \...
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Problem with deriving an expression of a parametric curve using Christoffel symbol.

Consider a particle moving in coodinate system $Z^i$. The parametric expression for the path of the particle is given by $Z^i=Z^i(t)$. The general component of the velocity vector is the time ...
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Are we allowed to define a symmetric (1,1) tensor in the following way?

I've recently stumbled accross the following task: "is it possible to define a symmetric and antisymmetric (1,1) tensor?". This is in context of relativity, so we have a metric $g$ at our ...
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Tensors: Meaning of the derivative terms in Covariant and Contravariant transformations.

I was reading some notes on tensors and was confused by a particular section which I shall write in bold. The Paragraph is as follows: In transforming between coordinate systems, a vector with ...
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Can a tensor's indices be flipped arbitrarily?

I made use of the following intermediate step in a demonstration I did: $$...=a_i\sigma_{kj}b_k=a_i\sigma^T_{jk}b_k=...$$ where the "T" just shows that the indices have been flipped. After ...

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