Questions tagged [kronecker-symbol]

For questions on kronecker symbols, a generalization of the Jacobi symbol to all integers.

Filter by
Sorted by
Tagged with
0 votes
1 answer
32 views

Unusual Kronecker Symbol

I am studying on an article about the Galerkin method and I found this symbole $\delta_{ij}^{km}$. I know the definition of the usual Kronecker Symbol which is : $$\delta_{ij}=\cases{1&if $i=j$\\...
Ada Az's user avatar
  • 79
0 votes
1 answer
43 views

How to make sense of a summation containing kronecker delta?

first question here so I hope it's appropriate. I'm looking at the following equation: And I'm struggling to figure it out. In the sum, there's a symbol that looks like a Kronecker delta (but that ...
os1's user avatar
  • 101
1 vote
1 answer
92 views

Evaluate a kronecker symbol sum: $\sum\limits_{n=1}^\infty \frac {\big(\frac n x\big)}n$ and $\sum\limits_{n=1}^\infty \frac {\big(\frac xn\big)}n$

The Kronecker Symbol $\left(\frac nm\right)$ has a range of $\{-1,0,1\}$ and $\sum\limits_{n=1}^\infty\frac{(-1)^n}n=-\ln(2)$, so we combine to find the following with the using software. Also note ...
Тyma Gaidash's user avatar
0 votes
1 answer
224 views

2 Levi Civita Summation [closed]

I dont understand how i can get the following expression: $$\varepsilon_{ijk}\varepsilon_{ijk'}=2\delta_{k k'}$$ I try the following and end up with a $-2$ instead of a $2$: $$\varepsilon_{ijk}\...
Beans9991's user avatar
0 votes
0 answers
115 views

How do I simplify $\delta_{ij} \delta^{jk}$?

How do I simplify $\delta_{ij} \delta^{jk}$? I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
math's user avatar
  • 93
-1 votes
1 answer
58 views

Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$

Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$ In the video, he directly cancelled $$3\delta_{jl}\delta_{im}-3\delta_{jl}\delta_{im}$$ and similar terms. I was thinking if subscript ...
user avatar
0 votes
1 answer
141 views

Difficult vector identity using Levi Civita

I have to prove the following: $$[3(\vec{p}\cdot\hat{r})\hat{r}-\vec{p}]\times[3(\vec{m}\cdot\hat{r})\hat{r})-\vec{m}]=-2\vec{p}\times\vec{m}+3\hat{r}[\hat{r}\cdot(\vec{p}\times\vec{m})]$$ I am given ...
Chris's user avatar
  • 2,946
0 votes
0 answers
112 views

Reducing tensor of rank two into Kronecker delta

I want to understand how this particular equality is true, and here we need to use the idea of contraction of tensors to achieve this theorem? $a_{ij}$ x $a^{ij} = \delta_j^j$ Here, a is a tensor of ...
Singularity's user avatar
-2 votes
1 answer
60 views

How to multiply a vector and a square matrix with Kronecker product, and know the answer's shape? [closed]

$\mathbf{1}_n \in \mathbb{I}^{n\times 1}$ is a vector of ones with shape $n\times 1$ $\mathbf{I}_m \in \mathbb{I}^{m\times m}$ is an identity matrix with shape $m\times m$ What is the answer to, and ...
develarist's user avatar
  • 1,514
0 votes
1 answer
42 views

Levi-Civta symbol question

$$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kl} (\delta_{jl}\delta_{ki} - \delta_{ji}\delta_{kl})$$ $$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kj}\delta_{ki} -\delta_{ii}\delta_{ji} = ...
Z. Huang's user avatar
2 votes
1 answer
935 views

Product of two Levi-Civita permutation symbols

I am attempting to follow the solution provided by (Mark Viola) here (Kronecker delta and Levi-Civita symbol). I do not understand the following: Step 1 & 2 where the initial Levi-Civita symbols ...
user825535's user avatar
2 votes
2 answers
308 views

name or symbol for "anti" Kronecker delta?

Is there a name or symbol convention for what I might call the "anti" Kronecker delta (that is, $1 - \delta_{ij}$)?
Grayscale's user avatar
  • 205
0 votes
1 answer
147 views

How to form probability density function using Kronecker symbol?

The question gives the PDF using two cases of random variable X. The solution begins with the PDF of X as stated below. I don't understand how they got to that point. I tried to take delta_0 to be 1 ...
Aurangzeb Rathore's user avatar
1 vote
2 answers
5k views

Prove that $\delta_{ij}\delta_{jk}=\delta_{ik}$

Firstly, I should mention that I have just started learning about tensors, the problem is that I need to understand why the result $$\fbox{$\delta_{ij}\delta_{jk}=\delta_{ik}$}\tag{1}$$ is true in ...
BLAZE's user avatar
  • 8,468
0 votes
1 answer
1k views

Taylor expansion using Kronecker tensor

I have the following function (just used as an example): $y_t=g(y_{t-1},\epsilon_t, \sigma)$ of which I have the following second-order Taylor expansion around a point such that $y=\bar{y}, \ \...
giorgio's user avatar
0 votes
2 answers
4k views

what is squared of a Kronecker ij?

Is it right to write $\delta_{ij}\delta_{ij}=(\delta_{ij})^2=\delta_{ij}$?
MathArt's user avatar
  • 1,090
0 votes
1 answer
58 views

How can I prove the following relation from tensor calculus?

$\frac{\partial \bar{x}_{i}}{\partial x_{r}} \frac{\partial {x}_{r}}{\partial \bar{ x_{j}}} = \delta^i_j \quad (The \quad Kronecker \quad Delta) \quad \quad\quad $ $\rightarrow ( \text{In my attempt, ...
Iaggo Capitanio's user avatar
2 votes
1 answer
578 views

Kronecker Delta Expressions

I am trying to understand the Kronecker Delta and want to clarify. Considering the definition of the Kronecker Delta and assuming $i=j=k$ for the following situations: I know that $\delta _j^i \delta ...
Cave Johnson's user avatar
2 votes
0 answers
286 views

Simplify bra-ket notation with kronecker product and kronecker sum

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation. I have the following equation(Swap gate with 3 cnot): First equation $|...
iconradez's user avatar
  • 143
1 vote
1 answer
2k views

Kronecker Delta with 3 indices

I want to express some equations in Einstein summation convention to improve readability and possibly simplify the calculations. I have searched for 3-dimensional versions of the Kronecker Delta, ...
Attack68's user avatar
  • 256
1 vote
0 answers
831 views

How to simplify notation for the Kronecker product of multiple matrices?

I would like to know if I can simplify notation for the Kronecker product of multiple matrices: $$A = A_1 \otimes A_2 \otimes \cdots \otimes A_n$$ For example, I could use a product sign such as: $$...
Igor Dakic's user avatar
2 votes
0 answers
40 views

Can I write $\frac {\partial E_{rs}} {\partial E_{mn}} = \delta_{rsmn}?$

Can I write $\frac {\partial E_{rs}} {\partial E_{mn}} = \delta_{rsmn}?$ and then use that delta to change variables in another tensor like: $$C_{ijkl}\delta_{rskl} = C_{ijrs}$$ or is there ...
Zduff's user avatar
  • 4,252
0 votes
1 answer
200 views

Show that the $n \times n$ identity matrix is commutative with any $n \times n$ martix using Suffix Notation

Using Suffix Notation, I have to show that the $n \times n$ identity matrix is commutative with any $n \times n$ martix with respect to matrix multiplication. We have just been introduced to the ...
LisaS's user avatar
  • 47
0 votes
1 answer
39 views

Compute $s_1 t_k \delta_{ii} \delta_{k1} \delta_{nn}$

I am in an argument with a friend from the university and we would like to clarify our problem: We have given the following term to calculate: $$s_1 t_k \delta_{ii} \delta_{k1} \delta_{nn}$$ All ...
Finn Eggers's user avatar
0 votes
0 answers
34 views

Calculate $δ_{ik}u_k$,$δ_{ij}δ_{ij}$

Maybe it's a dumb question, but I'm trying to understand some tensor/kronecker calculations that are something off-course : Calculate the expressions : $$δ_{ik}u_k,δ_{ij}δ_{ij}$$ So : $$δ_{ik}u_k = ...
Rebellos's user avatar
  • 21.3k
4 votes
2 answers
14k views

Proof of $\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$ [duplicate]

I'm a student of physics. There is an identity in tensor calculus involving Kronecker deltas ans Levi-Civita pseudo tensors is given by $$\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}...
SRS's user avatar
  • 1,452
3 votes
2 answers
2k views

Levi Civita and Kronecker Delta

I've been working on some quantum mechanics problems and arrived to this one where I have to deal with subscripts. I got stuck doing this: I have $\epsilon_{imk}\epsilon_{ikn}=\delta_{mk}\delta_{kn}-\...
RicardoP's user avatar
  • 215
1 vote
1 answer
961 views

Units of infinitessimal sum with kronecker delta when converted to integral

I'm having trouble understanding the following. Say I have a quantity of objects arranged spatially, with distribution $dn/dx$. For one reason or another*, I want to sum auto-pairs of spatial points, ...
StevenMurray's user avatar
4 votes
1 answer
8k views

Prove that $x\frac{d}{dx} \delta (x) = -\delta (x)$

Is this proof formal enough? I plan on being a theoretical physicist one day, so I want to get into the good habit of being mathematically strict. My proof: $u=x$; $du=dx$ $v = \delta (x)$; $dv = -\...
whatwhatwhat's user avatar
  • 1,587
3 votes
0 answers
10k views

Proof that Kronecker Delta is Mixed Tensor

The book I am reading asks the reader to verify that the Kronecker Delta is a second-order mixed tensor with one contravariant and one covariant index as indicated: $$ \delta_j^i = \left\{\begin{...
Tim Clark's user avatar
  • 418
1 vote
1 answer
201 views

Constructing an expression involving Kronecker delta and Levi-Civita symbol

Let $i,j,k,l\in\{0,1,2\}$. I am looking for a simple expression $f(i,j,k,l)$ involving only (Kronecker delta) $$\delta_{ab}=\cases{1&if $a=b$\\0&else},$$ and (Levi-Civita symbol) $$\epsilon_{...
Daniel Robert-Nicoud's user avatar
4 votes
2 answers
1k views

Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'...
Matt Miller's user avatar
2 votes
1 answer
1k views

Is there an opposite of the Kronecker Delta?

Instead of $\delta(n,n) = 1$ and $\delta (n,k) = 0$, is there something that returns $0$ when the arguments are the same, and $1$ when the arguments are different. Is there a special function that ...
jackzellweger's user avatar
2 votes
1 answer
114 views

Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
Jeesubmunu's user avatar
2 votes
0 answers
85 views

Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
snowman's user avatar
  • 3,733
1 vote
2 answers
1k views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) $\delta_{i\,j}...
Cara's user avatar
  • 11
1 vote
1 answer
106 views

A simple algebraic expression for Kronecker delta when both arguments take values 0 or 1

I am not sure whether this question makes sense but if it does and if it has some answer, then that would hugely simplify my task. I am looking for an algebraic expression for Kronecker delta $\delta_{...
Peaceful's user avatar
  • 531
0 votes
1 answer
1k views

Kronecker delta equation simplification

I am trying to simplify a tensor equation with Kronecker delta $$ A_{ij} \big ( \delta_{ik}\delta_{jm} -\frac{1}{3}\delta_{ij}\delta_{km} \big) $$ $A$ and $\delta$ are Cartesian tensors. I know ...
343_458's user avatar
  • 461
4 votes
2 answers
2k views

Why is the delta function the continuous generalization of the kronecker delta and not the identity function?

In a discrete $n$ dimensional vector space the Kronecker delta $\delta_{ij}$ is basically the $n \times n$ identity matrix. When generalizing from a discrete $n$ dimensional vector space to an ...
asmaier's user avatar
  • 2,660
1 vote
1 answer
121 views

A Summation Convention – Substitution Rule

I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian ...
stu_m's user avatar
  • 11
2 votes
1 answer
3k views

Integral definition of delta function and Kronecker symbol

I know the following two definitions for the delta function and Kronecker delta, respectively: (1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$ (2) $\int_{-\pi}^{\pi}\frac{e^{...
AlphaOmega's user avatar
2 votes
1 answer
229 views

Counting triples with a fixed sum using Kronecker delta and complex integration

Let $(n_1,n_2,n_3)$ a triple of non-negative integers summing up to $27$, i.e. $n_1+n_2+n_3=27$. I want to count how many triples there are satisfying this constraint, using the contour integral ...
Pierpaolo Vivo's user avatar
3 votes
1 answer
3k views

The Relation Between Kronecker's Delta and the Permutation Symbol

The Kronecker's Delta is defined as $$\delta_{ij}= \begin{cases} 1 & i=j \\ 0 & i \ne j \end{cases}$$ Also, the Permuation Symbol known as Levi Cevita's Symbol is introduced as $$\...
Hosein Rahnama's user avatar
0 votes
1 answer
2k views

Is $Q_p^q(\lambda) = (\delta_{ij} + \lambda\delta_{ip}\delta_{jq})$

Is this correct: $Q_p^q(\lambda) = (\delta_{ij} + \lambda\delta_{ip}\delta_{jq})$ With $Q_i^j(\lambda)$ is the unit matrix with $\lambda$ at the i-th row and j-th column (author calls it type II unit ...
Clayton Louden's user avatar
0 votes
0 answers
91 views

Subscript notation logic check

Given $$\nabla\cdot(\textbf{r}\times\nabla f)~=~(\nabla\times\textbf{r})\cdot\nabla f~-~\textbf{r}\cdot(\nabla\times\nabla f)$$ I would split the equation into 2 part: $(1)~~~~~~~~~(\nabla\times\...
Alana's user avatar
  • 193
1 vote
0 answers
88 views

Loop integral with Kronecker-delta notation

I worked through a problem and arrived at the final solution (which is correct), however, one part of it should equal zero mathematically. This is the part that should equal zero: $F_{2i}= (\frac{...
JadeChee's user avatar
6 votes
1 answer
1k views

Confusion about the Kronecker $\delta$

Something disturbs me, concerning the Kronecker $\delta$. Assuming these hold: $$\delta_{ij}\delta_{jk}=\delta_{ik}$$ $$\delta_{ij}=\delta_{ji}$$ $$\delta_{ii}=1$$ does it follow that for every $\...
Whyka's user avatar
  • 1,953
0 votes
1 answer
1k views

How to show that the isotropic tensor of order $n$ is a multiple of the Kronecker delta?

I have already found this question here but with the property of invariant under rotation. However, I don't have this property and I want to prove that $T_{ij} = \alpha \delta _{ij}$ where $T_{ij}$ ...
user189013's user avatar
0 votes
0 answers
582 views

Turning this double sum with Kronecker delta into a single sum - am I correct?

I would be grateful if you checked my solution: $$ \sum^{\lfloor j/2 \rfloor}_{q=0} \sum^{\lfloor k/2\rfloor}_{s=0} \frac{2^{k-2s}A^{q+s}}{q!s!(k-2s)!}\delta_{j-2q,k-2s} $$ Here $\lfloor \rfloor$ ...
Yuriy S's user avatar
  • 31.5k
3 votes
0 answers
97 views

Clarify the definition of a transition matrix

I am reading the book Matrix Variate Distribution by A. K. Gupta and D. K. Nagar. In the first chapter (Definition 1.2.8), they define a matrix $B_{p}$ ($p \in \mathbb{N}^{\ast}$) as follows : ...
Odile's user avatar
  • 1,171