Questions tagged [tensors]

For questions about tensors, 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.

Tensors, are arrays of numbers, or functions, that transform according to certain rules under a change of coordinates. In physics, tensors characterize the properties of a physical system.

A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. This just means that the tensor is defined at every point within a region of space (or space-time), rather than just at a point, or collection of isolated points.

Tensor is a geometric object that maps in a multi-linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Vectors and scalars which are often used in elementary physics and engineering applications, are considered as the simplest tensors. Vectors from the dual space of the vector space, which supplies the geometric vectors, are also included as tensors.

Tensors were conceived in $1900$ by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ... ) and others.

References:

https://en.wikipedia.org/wiki/Tensor

http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

https://www.quora.com/What-is-a-tensor

What are the Differences Between a Matrix and a Tensor?

An Introduction to Tensors

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Solving a tensor equation from singularity condition

I apologize if this question has already been asked but I'm not sure what the best key-words are. I have a tensor equation of the form: $(A + x\otimes b)c=0$. Here, $A$ is a second-order tensor/matrix, and the rest are first-order tensors/vectors.…
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On the effect of action of product of permutation group on vectors

I am confused with the following permutation. Suppose $\sigma,\tau\in S_n$ the permutation group of order $n$. Define $\sigma.(x_1,x_2,\dots, x_n)=(x_{\sigma(1)},x_{\sigma(2)},\dots, x_{\sigma(n)})$. (it affects on the indices directly, no matter…
C.F.G
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Questions on invariant tensors and representations of $SU(n)$ and $SO(n)$

I'm aware that the invariant/isotropic tensors in $SO(n)$ are $\epsilon^{i_1 i_2 \dots i_n}$ and $\delta^{ij}$. We can use the latter to find the trace of a tensor, Tr$(X)=\delta_{ij}X^{ij}$. Hence, we can decompose a general tensor into the…
martin
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Calculation of permutation in tensor of rank 4

I am trying to compute the symmetric part of a 4th order tensor $A_{ijkl}$ From a previous post (Symmetric Part of Product of 2 tank 2 tensors), I saw that I need to compute the permutations of $A_{ijkl}$ (4!=24 permutations). I would like to know…
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Commutativity of Contractions

I'm working through Pavel Grinfeld's book 'Introduction to Tensor Analysis and the Calculus of Moving Surfaces' and I am having trouble understanding the example given in section 4.9.2 - Commutativity of Contractions: For an illustration, consider…
D-Dᴙum
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Unconstrained quartic optimization

I am working on antenna array pattern synthesis algorithms, and am trying to minimize the following expression with respect to $\mathbf{v}$ $$ \int_\Omega \Big[ \big( \mathbf{v}^H \mathbf{Y}(\vartheta, \varphi) \mathbf{v} \big)^2 - \mathbf{v}^H…
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Is there a natural way to define the inverse of a tensor?

I have a second rank tensor $\mu$, which is in my case a linear map from $S^3(\mathbb{R})$ to $S^3(\mathbb{R})$ where $S^3(\mathbb{R})$ are the $3 \times 3$ symetric matrices with real coefficients. I was wondering if there exists a somehow "natural…
Velobos
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Parallel transport equation of a covariant component

I am working through Paul Renteln's book "Manifolds, Tensors, and Forms" (As I am learning General Relativity). I have come across a derivation of a 'parallel transport equation': $$\frac{d\gamma^i}{dt}\left(\frac{\partial Y^k}{\partial…
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Compute cartesian components of the curl

Given a vector $v$, the curl of $v$ is defined as the unique vector field with the property $$(\nabla v - \nabla v^T) a = (\text{curl } v) \times a$$ for every vector $a$. (See pag. 32 of Gurtin's book) I want to find its cartesian components (pg.…
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How does the Kronecker delta select coodinates?

I'm reading A Gentle Introduction to Tensors. In introducing the Kronecker delta it says it "[...] it may be used as a coordinate selector, in the following sense:" $\delta_j^i x^j = x^i$ Where superscript indices are rows, and subscript indices are…
yrom1
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Multiplication of rank 3 tensors to give another rank 3 tensor of the same dimensions

I have two rank three tensors, X and Y of the same dimensions, say n x n x n. I wish to do the equivalent of matrix multiplication on these two tensors. So with regular matrix multiplication, if I multiply two n x n matrices I get another n x n…
LOC
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$\sum_{i_{K+1},...,i_N=1}^N ε^{i_1,...,i_K,i_{K+1},...,i_N}δ^{j_{K+1},...j_N}_{i_{K+1},...,i_N}=(N-K)!ε^{i_1,...,i_K,j_{K+1},...,j_N}$

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 not read what is written in the image and so you…
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Order the basis of a Lie algebra so that multiplication is increasing?

Let $\frak{g}$ be an arbitrary Lie algebra over a field $K$ with a fixed basis $e_1,\ldots,e_m$. Is it possible to impose a total order $e_1<\ldots
Leo
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Do Tensors "act" on each other?

I know that , Derivatives are a kind of Linear Transformation. (Because they are a linear operator) Linear Transformation Are a kind of Tensor. So Are Derivatives just another kind of tensor? If that is , it seems like taking the Derivatives of…
Prithu Biswas
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Are there any non-discrete definitions for the size of a matrix?

Throughout my math education I have noticed that in order to solve a difficult problem with one set of numbers it helps to move to a larger encompassing set. For example, subtracting some natural numbers, $\mathbb{N}$ , requires the integers, …
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