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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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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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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
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
13 votes
1 answer
2k views

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 ...
Phibert's user avatar
  • 773
2 votes
1 answer
347 views

Mathematical properties of Rank-$N$ tensors where $N$>2

WARNING: This question might not have all the necessary tags. I asked about the uses of rank-$N$ tensors in physics on physics stackexchange, but for some reason it was closed saying that my question ...
Rounak Sarkar's user avatar
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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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
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
  • 7,441
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
3 votes
2 answers
584 views

Why is tensor from a vector space covariant, not contravariant?

My teacher defined a tensor as a linear application $$T: V \times V \times \dots \times V^* \dots \times V^* \rightarrow \mathbb{R} $$ (given $V$ vector space and $V^*$ its dual). After other things ...
fcoulomb's user avatar
  • 349
241 votes
8 answers
141k views

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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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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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
9 votes
2 answers
18k views

Levi Civita and Kronecker Delta identity [duplicate]

One of the popular Kronecker delta and Levi-Civita identities reads $$\epsilon_{ijk}\epsilon_{ilm}=\delta_{jl}\delta_{km}-\delta_{kl}\delta_{jm}.$$ Now, is there an intuition or mnemonic that you use, ...
Isomorphic's user avatar
  • 1,182
5 votes
1 answer
648 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
stebu92's user avatar
  • 388
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
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
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
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
8 votes
3 answers
3k views

Covariant derivative geometric interpretation

I'm having some trouble understanding what the covariant derivative means geometrically. I know the definition which states that for a tensor T with any number of indices: $ \nabla_j T = \frac{\...
glaba's user avatar
  • 422
7 votes
1 answer
5k views

Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
Blake's user avatar
  • 213
7 votes
2 answers
4k views

what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
ahala's user avatar
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6 votes
1 answer
832 views

Solving systems of quadratic equations

There are efficient algorithms for solving a system of linear equations of the form $$\forall i \qquad 0 = a^i + \sum_j b^i_j x^j$$ or $$\mathbf{0} = \mathbf{a} + \mathbf{b} \cdot \mathbf{x}$$ Are ...
user76284's user avatar
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3 votes
1 answer
3k views

Understanding the definition of norm of tensors on a Riemannian manifold

I am teaching myself Riemannian Geometry in order to studying Mean Curvature flow. I was reading Lecture Notes on Mean Curvature Flow by Carlo Mantegazza and I'm trying understand the following ...
George's user avatar
  • 3,837
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
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
38 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
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
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
  • 15.1k
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
  • 682
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
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
  • 1,175
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
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
12 votes
3 answers
3k views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
phizzy's user avatar
  • 121
9 votes
2 answers
5k views

Tensor Calculus

I am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III. I've started self studying tensor calculus, my ...
Salim's user avatar
  • 101
8 votes
2 answers
5k views

Finding a basis for symmetric $k$-tensors on $V$

We say a function is $k$-linear if it takes $k$ values as input and is linear with respect to each of them. For example, determinant is a $n$-linear function. (If the matrix is $n \times n$) A tensor ...
Arman Malekzadeh's user avatar
7 votes
2 answers
1k views

Why is enveloping algebra called enveloping algebra?

What does the enveloping algebra of $\mathfrak{g}$ have to do with envelopes? If $\mathfrak{g}$ is a Lie algebra, we take tensor algebra on $\mathfrak{g}$ and make quotient through ideal of T, so we ...
Tereza Tizkova's user avatar
7 votes
1 answer
9k views

Working out a concrete example of tensor product

From this entry in Wikipedia: The tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$. It is denoted $V\otimes_K W$, or $V\otimes W$ when the ...
Antoni Parellada's user avatar
7 votes
1 answer
24k views

What is the divergence of a matrix valued function?

According to Wikipedia: The divergence of a continuously differentiable tensor field $\underline{\underline{\epsilon}}$ is: $$\overrightarrow{\operatorname{div}}\,(\mathbf{\underline{\...
shinjin's user avatar
  • 381
6 votes
1 answer
22k views

Tensors and matrices multiplication

I have to prove an equality between matrices $R=OTDO$ where $R$ is a $M\times M$ matrix $O$ is a $2\times M$ matrix $T$ is a $M\times M\times M$ tensor $D$ is a diagonal $2\times 2$ matrix The ...
Augustin's user avatar
  • 8,486
6 votes
2 answers
2k views

What is the definition of tensor contraction?

According to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V^* \...
Jesse Madnick's user avatar
5 votes
2 answers
348 views

A doubt on Tensors: Can they be 1-form valued?

A I know a restrictive $-$ though sufficiently general in physics $-$ definition of the Tensor object: A $(p,q)-$Tensor is a multilinear function like: $$ T: V\times\cdot\cdot\cdot\times V\times V^{*}...
M.N.Raia's user avatar
  • 895
5 votes
1 answer
330 views

How to show $\nabla (\Delta A) -\Delta(\nabla A)=\nabla Rm *A+Rm*\nabla A$?

$A$ is a tensor field, $\nabla A(X,...)=(\nabla_XA)(...)$. Besides, $R$ is curvature operator and $$ Rm(X,Y,Z,W)= \langle R(X,Y)Z,W\rangle $$ and $$ \Delta A = g^{ij}(\nabla _{X_i}\nabla _{X_j}A - \...
Enhao Lan's user avatar
  • 5,895
5 votes
1 answer
326 views

How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and $\...
user111187's user avatar
  • 5,846
5 votes
2 answers
312 views

How to understand $R^{\rho}_{\sigma\mu\nu}$ for constructing geometries?

I'm trying to understand the curvature tensor $R^{\rho}_{\sigma\mu\nu}$ by playing with it in certain contexts. I already have an understanding, to some degree, of how it measures the change in a ...
Alexander Conrad's user avatar
4 votes
1 answer
938 views

Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
ahala's user avatar
  • 3,050
4 votes
1 answer
2k views

Deriving Ricci identity for co-vector fields

Let $\nabla$ be the covariant derivative associated with a torsionless connection. Prove the Ricci identity for covectors: $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\...
CAF's user avatar
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