Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [tensors]

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 transforms under certain rules with coordinate changes). For questions about the more abstract aspects of tensor products, use the (tensor-products) tag instead.

-1
votes
0answers
20 views

Direct sum and normal sum [on hold]

What is the differences between $(\hat{e_1}+2\hat{e_2})(e_1+e_2)$ and $(\hat{e_1}\oplus 2\hat{e_2})(e_1+e_2,e_2)\\$ $e_1=(1,0,0), e_2=(0,1,0)$
-1
votes
0answers
12 views

Help with proving Angular Momentum Balance and Symmetry of stress tensor [on hold]

I can't for the life of me figure this out. I have pored through two textbooks on transport phenomena, and this isn't making any sense. Nothing online seems to have a good suggestion for how to solve ...
0
votes
1answer
18 views

Which indices go where in the resulting delta function

I want to understand what the following is equal to in terms of delta functions: $\dfrac{\partial(\partial_\alpha A_{\beta})}{\partial(\partial_\mu A_\nu)}$ I know this is equivalent to: $\dfrac{\...
0
votes
0answers
14 views

Proof of Apolarity Lemma [on hold]

I’m trying to prove Apolarity Lemma but I have some problem in proving the non trivial implication. Thanks for your help.
2
votes
0answers
45 views

What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?

Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation $$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$ where $\nabla\mathbf{u}$ ...
2
votes
1answer
34 views

Prove that the covariant derivative commutes with musical isomorphisms

Suppose I have a covector field $\omega$ and a covariant derivative $\nabla_{X}$ for some vector field $X$ on a Riemannian manifold $(M, g)$. Define $X^{\flat} \in \mathfrak{X}^{*}(M)$ as $X^{\flat}(...
0
votes
1answer
16 views

Simplification of multiple Levi-Civita Epsilons

Given the expression: $$\epsilon_{ijk}\epsilon_{klm}\epsilon_{mni}=c\epsilon_{jln}$$ where $c$ is a constant I have to determine. Given the fact: $$\epsilon_{jln}\epsilon_{jln}=6$$ I have said $$c=\...
0
votes
1answer
26 views

Prove Poincaré Lemma for $1$-form

Let $U\subseteq\mathbb{R}^n$ be an open set that contains $0$, and for all $t\in[0,1]$ and $ x\in U$, $tx\in\mathbb{R}^n$. Show that every closed differentiable 1-form $w$, (i.e. $dw=0$) is an exact ...
0
votes
0answers
23 views

How do you write a x b + c = (a.b)b-d in suffix notation? [on hold]

Struggling to rewrite a few things in suffix notation, any help is welcome. Thanks.
0
votes
1answer
16 views

Simplification using suffix notation

I have a question and I need to simplify $(a\times b)+c=(a\cdot b)b-d$ using suffix notation. $(a\times b)_i=\epsilon_{ijk}a_jb_k$ and $(a\cdot b)_i=a_jb_j$ So far I have got to $\epsilon_{ijk}...
0
votes
0answers
9 views

Basic resources for learning tensors

What are some "easy" resources /intro to tensors? (tensor products etc.). I am taking intro to QM right now and the professor is just telling us how to do basic tensor products on kets, operators ...
1
vote
0answers
39 views

Question on Partial derivatives in inverse function changing from coordinate systems

I would like to ask a question about partial derivatives in the context of Rotations of coordinate systems. Say we have a coordinate system (unprimed) and its rotated version (primed). If the ...
-1
votes
0answers
29 views

Tensor product identities

I have to prove the following: $$S(a⊗b) = (Sa)⊗b$$ $$(a⊗b)S=a⊗(S^Tb)$$ $$(a⊗b)^T=b⊗a$$ $$tr (a⊗b) =a·b$$ $$(a⊗b) :S= (Sb)·a$$ I'm new with tensor calculus and my teacher ...
0
votes
1answer
22 views

Tensor contraction computation

I am currently trying to implement tensors as multidimensional arrays in C++, which is why i am asking this with regards to algorithmic computability. I would like to implement the contraction of two ...
2
votes
1answer
33 views

Why is the volume element unique in the way Spivak develops it?

In Spivak's Calculus on Manifolds, he develops the volume element in the following way: The fact that $\dim \Lambda^n(\mathbb{R}^n) = 1$ is probably not new to you, since det is often defined as ...
0
votes
0answers
22 views

Tensor cross product proof: $Sa \times Sb = \det(S)S^{-T}(a \times b)$ [duplicate]

I need to prove $$Sa \times Sb = \det(S)S^{-T}(a \times b),$$ given that $a$ and $b$ are vectors and $S$ is a second-order (rank $2$) tensor. I have the hint: vectors $u=v$ iff $u\cdot a=v\cdot ...
0
votes
0answers
17 views

Tensor and the $k$ times derivative of a smooth function $f$

I am currently taking an university analysis class. I learned yesterday in the class that (1) if $f: \mathbb{R^n} \rightarrow \mathbb{R}$ is smooth, then $D^kf: \mathbb{R^n} \rightarrow T^k(\mathbb{R^...
0
votes
2answers
54 views

What is the coordinate-free definition of the Levi-Civita Symbol?

For clarity I mean the symbol $\epsilon_{ijk}=e_{ijk}/|g|$ where $g$ is the determinant of the metric tensor and $e_{ijk}$ is anti-symmetric in all its indices and $e_{123}=1$. Is there a purely ...
0
votes
1answer
21 views

A difficulty in understanding the indices in the matrix interpretation of the product of two representation.

I have a difficulty the indices in the two lines above the line starting with "Hence, $X$ transforms according to ..." , could anyone explain for me how the indices are located? Also I do not ...
1
vote
1answer
28 views

How to use operators on tensor products

This problem is from physics, but I have trouble understanding the math. I have some problem understanding how to use tensors. Let's say in Quantum Optics if I have the state in mode $b$ (where I can ...
1
vote
1answer
15 views

deriving the Poisson equation from Navier stokes equation using tensor algebra.

I want to derive the poisson equation from the navier equations, but I'm not able to do this. I'm able to substitute the index form in place of the operators in the poisson equation and get back ...
0
votes
0answers
34 views

curvature tensor of involutive distributions

The Riemann curvature tensor is defined $$R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z.$$ Is it possible to simplify above expression under assumption that $X,Y\in D$, where $D$ ...
0
votes
2answers
32 views

Time derivative of scalar function that takes vector as argument

Let's say I have scalar function $\phi$ that is function of some vectors $\vec{\bf{p}}$ and $\vec{\bf{r}}$ such that $\phi = \phi(\vec{\bf{p}}-\vec{\bf{r}})$, also vector $\bf{r}$ is function of time, ...
0
votes
0answers
10 views

Derivative of Square of Covariant Derivative of Riemann Tensor

I am reading some notes on Ricci flow where it is stated that $\frac{\partial}{\partial t}|\nabla Rm|^2 \leq \Delta|\nabla Rm|^2 - 2 |\nabla^2 Rm|^2 + C |Rm||\nabla Rm|^2$. The text states that you ...
0
votes
0answers
20 views

What's a good book about tensors

I've tried learning basic tensor calculus (transformation law, inner & outer products, adding , etc..) and I could 'work' with the math with no problem, but sometimes I couldn't quite understand ...
0
votes
0answers
11 views

Explicit form of elements of tensor products

Let $V$ and $W$ be two vector spaces over $\mathbb{F}$. We know that their tensor product $V \otimes W$ is also a vector space over $\mathbb{F}$. I am wondering how would look like the elements inside ...
4
votes
2answers
139 views

Covariant derivative of a symmetric tensor

Assume that a symmetric $(0,2)$ satisfies $$\nabla_iT_{jk}+\nabla_jT_{ik}+\nabla_kT_{ji}=0$$ where $T=T_i^i$ is constant and $\nabla_jT_{ik}\ne 0$. What are the values of the constants $a,b,c$ such ...
0
votes
0answers
14 views

divergence theorem for a rank 2 tensor

I wanted to double-check on something. We all know that for a vector field $T$: $\int dV \nabla \cdot T = \int d\hat{n}\cdot T$ My understanding is that this is still true if $T$ is a rank-2 tensor,...
3
votes
0answers
59 views

Upper bound CP tensor rank

I have a question about CP tensor ranks. In the following, $\mathcal X \in \mathbb R^{n_1 \times n_2 \times n_3}$ is a third-order tensor of CP rank $R$, i.e., there exist vectors $a_i$, $b_i$ and $...
0
votes
0answers
25 views

Tensor methods for representations of SU(n)

I'm familiar with the Young Tableaux method for finding irreducible representations of $\mathrm{SU}(n)$ for $n\in\mathbb{N}$, and I also know of the tensor methods for finding irreducible ...
0
votes
0answers
29 views

tensor transformation

I don't understand a passage about tensor transformations in the book I'm reading. A tensor obeys the transformation law: If $A_{\mu\nu}' = b_{\mu \alpha }b_{\tau \alpha } A_{\mu\nu} $ Where the $b_{...
0
votes
0answers
29 views

Formula of the gradient of vector dot product

On Wikipedia in the article "Vector calculus identities" (https://en.m.wikipedia.org/wiki/Vector_calculus_identities) there are the following two formulas for computing the gradient of vector dot ...
0
votes
0answers
24 views

A special construction in $\mathbb{R}^n$

I have a question from my professor' notes. We defined $\Lambda^k(V)$ as the set of all $k$-Tensor' forms (multilinear transformations), $\omega$, which fulfuill $\omega(v_1,...,v_i,v_j,...,v_k)=-\...
0
votes
0answers
12 views

Maxwell-Chern-Simons equations: Translating from form language to component form

I am trying to solve the scalar-coupled Maxwell-CS equations, which is written in this form in the differential form language \begin{eqnarray} 0=d(Q_{IJ}\star F^J)+\frac{1}{4}C_{IJK}F^J\wedge F^K \...
0
votes
0answers
22 views

Integrating with respect to a tensor

Without loss of generality, Let's say we have a tensor $A$ of order 3 and a differential form $dV$ of order 2. Then how would we evaluate the following indefinite integral? $$I=\int A dV$$. My first ...
0
votes
1answer
40 views

When is the nth component of a (co)vector equal to its scalar product with the nth element of its dual basis?

My solution to a problem in a tensors book is different from the solution in the book, and I don't know why. Here's the problem: $$\vec e_1 = (2, 1) \hspace{1em} \vec e_2 = (-1, 3) \\ \text{Find the ...
0
votes
0answers
25 views

Why would the following define a 1-1 tensor?

Let $g$ be a Riemannian metric on a manifold $M$, and let $\mu_g$ be the volume element, i.e. for any $x \in M, v,w \in T_x M$, we have $$\mu_g(v,w)=\pm\sqrt{\det\begin{bmatrix} g(x)(v,v) & g(x)(v,...
0
votes
0answers
16 views

Is Multi-dimensional arrays always enough information to describe a Tensor?

Given a vector space V with finite independent basis Q containing D elements. $V^*$ denotes the dual space, and $V^*$ has also independent basis E. Linear combination of elements in Q can represent ...
2
votes
1answer
27 views

Can you take the dot product of a column vector and a row vector (i.e. a vector and a dual vector)

I recently learned about the definition of work, namely the one involving path integration. W=integral of (F.dr). In this case, F is a vector field and dr is a small segment of the path r being ...
1
vote
3answers
37 views

Why can we contract an upper index with only a lower index?

Why can't we contract two upper or two lower indices? Can you explain this with the help of matrices? Moreover, how can we represent the operation of a (0,2) tensor on vectors with a matrix?
2
votes
1answer
52 views

How do I tell the rank of the electric susceptibility tensor (and others)?

I understand that a tensor is a multilinear map from $V^*\times\cdots\times V^*\times V\times\cdots\times V$ to $V$'s underlying field, where $V$ is a vector space and $V^*$ its dual. This is fine, ...
1
vote
3answers
60 views

Tensor equation [closed]

If you have two antisymmetric tensors $A_{\mu \nu}$ and $B_{\mu \nu}$, and for every anti symmetric tensor $\epsilon^{\mu \nu}$, $\epsilon^{\mu \nu} A_{\mu \nu} = \epsilon^{\mu \nu} B_{\mu \nu}$ Is ...
1
vote
1answer
24 views

Proving zero identities in vector calculus with simple arguments involving index counting or symmetry?

Consider the following table describing four second derivative operators. ...
0
votes
1answer
14 views

How do I write this multiple index tensor equation using Ricci calculus where the 4-gradient is acting on each tensor?

In the context of special relativity I have this to show that this equation is correct: $$\partial_\mu F^{\mu \nu}=j^\nu$$ To do that I'm trying to take this equation: $$F^{\mu \nu}=\partial^{\mu}A^{\...
0
votes
0answers
13 views

product space, Tensors( Dyad), Kronecker Delta, metric tensor

I have four questions -- It is mentioned that an n-dimensional space and an m-dimensional space may be used to determine a new and unique (n+m)-dimensional product space. if we consider two circles ...
0
votes
0answers
21 views

Display the values of Christoffel symbols in simplified form in Maxima software

I have calculated all Christoffels symbols (my metric is symmetric type) by hand and now I am trying a lot to compute all of the non-zero values of Christoffel symbols by maxima just for confirmation ...
0
votes
2answers
44 views

Tensor Dot Product of two tensors of arbitrary order

I am currently working on implementing the inner(scalar or dot) product of two tensors of arbitrary order. As far as i understand, you need to make sure, that the last dimension of the first tensor $\...
1
vote
1answer
35 views

How to optimize a non-linear least squares energy with respect to the non-zeros of a sparse matrix?

I have an energy I'd like to minimize of the form: $E(G) = \|\underbrace{X - Y G^T L G B}_{f(G)}\|_F^2$ where $X,Y,B$ are dense matrices and $G,L$ are sparse matrices ($G^TLG$ is also sparse), $\|M\...
1
vote
1answer
48 views

What's the geometric explanation of Lie Derivative and Covariant Derivative?

What's the geometric explanation of Lie Derivative and Covariant Derivative? For my understanding, covariant derivative somehow explain/ provided a way for determine the invariance or for coordinate ...
2
votes
1answer
44 views

covariant metric tensor in case of polar transformation

Assume usual polar coordinates equations: $$ x=r\cos(\theta) $$ $$ y=r\sin(\theta) $$ In what I think is called covariant we have a Jacobian of: $$ J=\begin{pmatrix} \frac{\partial{x}}{\partial{r}} ...