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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.

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What are the Differences Between a Matrix and a Tensor?

What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
Aurelius's user avatar
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180 votes
6 answers
35k views

An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
Noldorin's user avatar
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158 votes
7 answers
19k views

Does a "cubic" matrix exist?

Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please don'...
Ian Mateus's user avatar
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94 votes
5 answers
52k views

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
Mike Flynn's user avatar
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72 votes
4 answers
40k views

Are there any differences between tensors and multidimensional arrays?

I see lots of references saying things like A tensor is a multidimensional or N-way array But others that say things like it should be remarked that other mathematical entities occur in ...
rhombidodecahedron's user avatar
47 votes
2 answers
21k views

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
Manikanta Borah's user avatar
43 votes
3 answers
32k views

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $...
Jānis Lazovskis's user avatar
43 votes
6 answers
9k views

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
user avatar
42 votes
2 answers
3k views

Is it misleading to think of rank-2 tensors as matrices?

Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation ...
user avatar
39 votes
6 answers
20k views

How to intuitively understand parallel transport

In the article I've referenced below, and many other articles for that matter, the notion of parallel transport along a line of latitude $\theta=\theta_0$ on the unit 2-sphere is spoken about. What I ...
Arturo don Juan's user avatar
38 votes
5 answers
42k views

Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
crasic's user avatar
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33 votes
5 answers
15k views

Why isn't there a contravariant derivative? (Or why are all derivatives covariant?)

Question: If there exists a covariant derivative, then why doesn't there also exist a "contravariant derivative"? Why are all or most forms of differentiation "covariant", or rather why do all or most ...
Chill2Macht's user avatar
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29 votes
2 answers
24k views

Prove that Christoffel symbols transformation law via the metric tensor

It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is: $$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ \...
Arthur's user avatar
  • 389
28 votes
3 answers
11k views

Why is a linear transformation a $(1,1)$ tensor?

Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and $...
Quantum spaghettification's user avatar
27 votes
3 answers
29k views

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's user avatar
27 votes
3 answers
7k views

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 ...
Hui Yu's user avatar
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25 votes
7 answers
11k views

How would you explain a tensor to a computer scientist?

How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
closedvolumeintegral's user avatar
25 votes
4 answers
3k views

Are the Ricci and Scalar curvatures the only "interesting" tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
Jesse Madnick's user avatar
24 votes
9 answers
4k views

Tensors in the context of engineering mechanics: can they be explained in an intuitive way?

I've spent a few weeks scouring the internet for a an explanation of tensors in the context of engineering mechanics. You know, the ones every engineering student know and love (stress, strain, etc.). ...
RobbieFresh's user avatar
23 votes
2 answers
13k views

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 ...
SDV's user avatar
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23 votes
1 answer
2k views

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....
Antoni Parellada's user avatar
23 votes
0 answers
613 views

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 ...
Ali Taghavi's user avatar
22 votes
3 answers
14k views

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 ...
HansHarhoff's user avatar
22 votes
1 answer
3k views

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 ...
caffeinemachine's user avatar
22 votes
6 answers
13k views

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. ...
kennytm's user avatar
  • 7,555
21 votes
3 answers
23k views

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 ...
user16186's user avatar
  • 541
21 votes
3 answers
41k views

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 ...
Sahil Chaudhary's user avatar
20 votes
2 answers
2k views

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....
KalEl's user avatar
  • 3,327
19 votes
2 answers
6k views

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 ...
user224772's user avatar
19 votes
1 answer
11k views

The Dimension of the Symmetric $k$-tensors

I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the ...
user20353's user avatar
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19 votes
6 answers
9k views

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 ...
Antoni Parellada's user avatar
19 votes
1 answer
13k views

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}+\...
Mark's user avatar
  • 197
19 votes
3 answers
7k views

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 ...
user avatar
18 votes
6 answers
10k views

Tensor Book Recommendation Request

Requirements Tensors Intuitive + Practical Reason for Tensor Introduction Current Knowledge Course Notes Abstract + Theoretical
18 votes
1 answer
3k views

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 ...
Pedro Lauridsen Ribeiro's user avatar
18 votes
3 answers
2k views

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 ...
Yrogirg's user avatar
  • 3,669
18 votes
1 answer
18k views

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^{...
user227550's user avatar
17 votes
4 answers
8k views

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 $q$ ...
user avatar
16 votes
2 answers
5k views

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, ...
Daniel Mahler's user avatar
15 votes
3 answers
6k views

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 ...
Serkan Yaray's user avatar
15 votes
2 answers
6k views

Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as ...
sifsa's user avatar
  • 969
15 votes
2 answers
17k views

tensor rotation

why does tensor rotation require multiplication by the rotation matrix twice, once from the right and once from the left by the inverse? if $T$ is the tensor I wish to rotate and $R$ is the rotation ...
Rubenz's user avatar
  • 421
15 votes
1 answer
3k views

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 ...
11Kilobytes's user avatar
  • 1,099
15 votes
1 answer
26k views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\...
Argyll's user avatar
  • 866
15 votes
2 answers
586 views

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 ...
ITA's user avatar
  • 1,823
14 votes
2 answers
5k views

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 ...
jaysonpowers's user avatar
14 votes
1 answer
733 views

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 $\...
Travis Willse's user avatar
13 votes
2 answers
2k views

Index notation for tensors: is the spacing important?

While reading physics textbooks I often come across notation like this; $$J_{\alpha}{}^{\beta},\ \Gamma_{\alpha \beta}{}^{\gamma}, K^\alpha{}_{\beta}.$$ Notice the spacing in indices. I don't ...
Giuseppe Negro's user avatar
13 votes
3 answers
2k views

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 ...
Life_student's user avatar
13 votes
3 answers
402 views

Show that $\left(\nabla_c\nabla_d-\nabla_d\nabla_c\right)v^a=R_{bcd}^av^b$ for vector field, $v$, and the Riemann curvature tensor, $R_{bcd}^a$

I'm going to be asking about parts of the author's solution to the following question. I feel this is a very important question to understand as it touches on some crucial aspects of the covariant ...
FutureCop's user avatar
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