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

3201 questions
21
votes
2 answers

Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since I couldn't find a similar question here on MSE,…
SDV
  • 632
  • 1
  • 4
  • 12
21
votes
3 answers

Tensors, what should I learn before?

Here I will be just posting a simple questions. I know about vectors but now I want to know about tensors. In a physics class I was told that scalars are tensors of rank 0 and vectors are tensors of rank 1. Now what will be a tensor of rank…
user16186
  • 431
  • 2
  • 5
  • 8
21
votes
3 answers

A user's guide to Penrose graphical notation?

Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation. The several references, especially ones with Penrose as…
Hui Yu
  • 14,131
  • 4
  • 33
  • 97
21
votes
1 answer

Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the Wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite dimensional vector space over a field $F$ and define a…
caffeinemachine
  • 16,806
  • 6
  • 27
  • 107
21
votes
0 answers

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each fibre(tangent space) gives a simple Lie…
Ali Taghavi
  • 1,740
  • 1
  • 11
  • 34
20
votes
5 answers

How to visualize a rank-2 tensor?

The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc. I know mathematically a tensor can be represented by a $3 \times 3$ matrix. But I can't grasp its geometrical picture —…
kennytm
  • 7,335
  • 4
  • 36
  • 37
20
votes
2 answers

Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation

For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the determinant. Why can the antisymmetrizing bar be inserted in…
KalEl
  • 3,219
  • 3
  • 17
  • 24
20
votes
3 answers

Are matrices rank 2 tensors?

I know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular or something else? Some sources I read seem to…
HansHarhoff
  • 303
  • 1
  • 2
  • 6
19
votes
3 answers

Multiplying 3D matrix

I was wondering if it is possible to multiply a 3D matrix (say a cube $n\times n\times n$) to a matrix of dimension $n\times 1$? If yes, then how. Maybe you can suggest some resources which I can read to do this. Thanks!
Sahil Chaudhary
  • 343
  • 1
  • 3
  • 10
19
votes
1 answer

Oh Times, $\otimes$ in linear algebra and tensors

Can I have some clarification of the different meanings of $\otimes$ as in the unifying and separating implications in basic linear algebra and tensors? Here is some of the overloading of this symbol... 1.1. Kronecker matrix product: If $A$ is an $m…
Antoni Parellada
  • 7,767
  • 5
  • 30
  • 100
19
votes
3 answers

Derivative of a vector with respect to a matrix

let $W$ be a $n\times m$ matrix and $\textbf{x}$ be a $m\times1$ vector. How do we calculate the following then? $$\frac{dW\textbf{x}}{dW}$$ Thanks in advance.
arindam mitra
  • 1,361
  • 1
  • 10
  • 17
19
votes
2 answers

What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both computationally and geometrically), but what does shear…
user224772
  • 191
  • 1
  • 5
18
votes
6 answers

Tensor Book Recommendation Request

Requirements Tensors Intuitive + Practical Reason for Tensor Introduction Current Knowledge Course Notes Abstract + Theoretical
17
votes
1 answer

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor calculus is tensor contraction, which can be…
17
votes
2 answers

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text d \boldsymbol s$ (vector) equals $\text d…