Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

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42
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5answers
5k 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, ...
89
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6answers
4k 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 ...
4
votes
1answer
149 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. $g^{ij}$ ...
12
votes
2answers
641 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 ...
15
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3answers
2k 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 ...
6
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3answers
2k 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 o and vectors are tensors of ...
5
votes
3answers
513 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 $q$ ...
5
votes
2answers
265 views

Unique symmetric covariant $k$-tensor satisfying $(\operatorname{Sym} T)(A,…,A)=T(A,…,A)$ for all $A \in V$

Let $T$ be a covariant $k$-tensor on a finite dimensional vector space $V$. I want to prove that the symmetrization of $T$ is the unique symmetric $k$-tensor satisfying the following condition: ...
6
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2answers
2k views

Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
4
votes
1answer
200 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
6
votes
2answers
314 views

Special case of the Hodge decomposition theorem

I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
10
votes
5answers
3k 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 3x3 matrix. But I ...
5
votes
1answer
203 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
4
votes
2answers
271 views

$\det(A \otimes B - B \otimes A) = 0$ why? Why $rk(M) = n^2-n$ ? Why x and -x in Spec(M) ?

Let $A$, $B$ be $n\times n$ matrices. It seems $\det(A \otimes B - B \otimes A) = 0$. Moreover it seems that the kernel of $A \otimes B - B \otimes A$ contains $n$ vectors. Here is MatLab code to ...
2
votes
1answer
303 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
6
votes
1answer
387 views

I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?

I am reading through Fung and Tong's "Classical and Computational Solid Mechanics", and feel that the Einstein summation convention saves a few symbols, at the expense of a lot of clarity. Along with ...
5
votes
2answers
600 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^* ...
2
votes
1answer
113 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
0
votes
0answers
39 views

Show that the tensor product is followed by a rotation and a dilation?

I am thinking mjqxxxx's sentence (a⊗b) [is] a projection followed by a rotation and dilation. We know that \begin{equation} (\textbf{a} \otimes \textbf{b}) \textbf{c} = (\textbf{b} \cdot ...
0
votes
1answer
2k 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: ...
4
votes
1answer
156 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
3
votes
1answer
149 views

Difference between tensor and tensor field?

I couldn't get 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 ...
2
votes
1answer
84 views

Where is the tensor product of two unit vectors projection onto?

I know that $\bar{e} \otimes \bar{e}$ is a projection onto $\bar{e}$. Then, I start to think where is then $\bar{e}_{i} \otimes \bar{e}_{j}$ projection onto. Where is the expression $\bar{e}_{i} ...
2
votes
2answers
319 views

Tensors of order 3

I'm wondering what a tensor of order 3 looks like, and what it's purposes are. I've seen them written down before, but they look like matrices; I'm probably not understanding the concept well. How is ...
1
vote
1answer
18 views

When is a symmetric 2-tensor field globally diagonalizable?

Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$. At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in ...
1
vote
1answer
76 views

Trace of tensor product

I am currently struggeling with the following problem: Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot ...
1
vote
1answer
77 views

Represent in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$

How can I represent this in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$ where all the entries are real and $W$ is a known(constant) matrix and $F$ is a rectangular matrix. When I say matrix ...
1
vote
1answer
252 views

How to solve a tensor differential equation?

Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$ where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor. The original Problem How does ...
1
vote
1answer
394 views

How to generate the inverse of a order 3 tensor

Is it possible to generate an inverse of an order 3 tensor? If so, how? I have been searching for a couple days, and cannot seem to find anything online to help with this.
1
vote
1answer
473 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 ...
1
vote
2answers
449 views

is there a way to solve the following tensor equation?

I have the following tensor (takes a vector of length $m$ and returns a matrix $m \times m$): $C(y) = A \operatorname{diag}(A^T y ) A^{-1}$ for some invertible matrix $A$ of size $m \times m$ ($y$ ...
0
votes
1answer
78 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...