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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Is there a fundamental problem with extending matrix concepts to tensors?

We are familiar with the theory of matrices, more specifically their eigen-theorems and associated decompositions. Indeed singular value decomposition generalizes the spectral theorem for arbitrary matrices, not just square ones. Now it only seems…
ITA
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Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various combinations of their tensor products. While it is easy…
user54031
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Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ vectors to $\mathbb{R}$. While generalizing to…
user23238
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How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field $\mathbb{F}$, and say (for lack of a better term) that the wedge…
Travis Willse
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Difference between the Jacobian matrix and the metric tensor

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial e_j}+\dfrac{\partial y}{\partial e_i}\dfrac{\partial…
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Do I understand metric tensor correctly?

So I've been studying vectors and tensors, and I'm trying to understand metric tensors. As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors regardless of whether their basis has changed. So…
jaysonpowers
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Determinant of a tensor

Is there such a thing as the determinant of a tensor of rank $\gt 2$? I am tried to think how it might be defined -- potentially like, the determinant of the tensor $A=a_{ijk}$ is $\det(A)=\epsilon^{ijk}\epsilon^{lmn}a_{1il}a_{2jm}a_{3kn}$. But I…
user227550
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How can I derive the back propagation formula in a more elegant way?

When you compute the gradient of the cost function of a neural network with respect to its weights, as I currently understand it, you can only do it by computing the partial derivative of the cost function with respect to each one of the weights…
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What is the practical difference between abstract index notation and "ordinary" index notation

I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, I am not clear on what practical difference this…
Daniel Mahler
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Vorticity equation in index notation (curl of Navier-Stokes equation)

I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation: $$ {\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = (\mathbf{u}\cdot\nabla)\pmb\omega - ( \pmb\omega…
Casio
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Why some operations on tensors don't give a tensor?

The gradient is a tensor $\nabla f:\mathbf{V} \to \mathbf{R}$ where the partial derivatives are evaluated at some point $(x_0, y_0, z_0)$. And evaluation of this linear form at some vector $v=(v_1,v_2,v_3)$ gives $$ (\nabla f)(\mathbf{v}) =…
user782220
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Difference Between Tensor and Tensor field?

I don't understand the difference between tensor and tensor field. I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions: If $A:(V^*)^r \times V^s\to K$ transformation is $K$-multilinear then $A$ is a tensor on…
Serkan Yaray
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Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical mechanics course, and within that context, I never…
Life_student
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Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is a tensor. So I realise the aim is probably to show…
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Understanding the definition of tensors as multilinear maps

The question arises from the definition of the space of $(p,q)$ tensors as the set of multilinear maps from the Cartesian product of elements of a vector space and its dual onto the field, equipped with addition and s-multiplication rules, given in…
Antoni Parellada
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