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Questions tagged [kronecker-delta]

For questions about the Kronecker-delta, which is a function of two variables (usually non-negative integers).

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Understanding Kronecker delta symbol & summation in 3D integral

I'm currently trying to practice finding the inertia tensor for simple rigid bodies, with the inertia tensor elements given by: $$I_{ij}=\int_{V}^{}\rho(\delta_{ij}\sum_{k}^{}x_{k}^2-x_{i}x_{j})dv$$ I ...
OldWorldBlues's user avatar
3 votes
1 answer
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Kronecker delta expressed as a derivative when there are multiple indices.

For instance, when differentiating four-vectors the result is straightforward: $$\frac{\partial x^\mu}{\partial x^\nu}=\delta_\nu^\mu$$ as the derivative is only non-zero when the Lorentz indices ...
digital's user avatar
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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
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3 votes
1 answer
112 views

Two similar proofs that $\frac{\partial}{\partial x'^\mu}={\Lambda_\mu}^\nu\frac{\partial}{\partial x^\nu}$, which one is correct?

Using the chain rule, show that the derivative transforms as $$\frac{\partial}{\partial x^\mu}\to\frac{\partial}{\partial x'^\mu}={\Lambda_\mu}^\nu\frac{\partial}{\partial x^\nu}\tag{A}$$ This is the ...
Sirius Black's user avatar
1 vote
2 answers
71 views

Sum with multiple Kronecker deltas

I have a problem dealing with the following expression: $$\sum_{i,j,m}\delta_{ij}\delta_{mj}\delta_{jm}x_ip_j$$ which I know it should yield the following result: $$ 3\mathbf{x}\cdot\mathbf{p} $$ I ...
Claudio Menchinelli's user avatar
-1 votes
1 answer
80 views

Why does the summation symbol disappear?

We know the orthogonality condition $$\int_0^1\sin(n\pi x)\sin(m\pi x)dx = \begin{cases} 0 & \text{ if } n \neq m\\ \frac12 & \text{ if } n = m\\ \end{cases} $$ From earlier in the text we ...
Gabsmacked's user avatar
1 vote
1 answer
45 views

Find a matrix such that the quadratic form of a orthonormal basis is equal to the Kronecker delta

For $n \in \mathbb{N}$ let $\{v_1, v_2, \ldots v_n \}$ be a orthonormal basis of $\mathbb{R}^{n}$. Further, let $i \in \{ 1, 2, \ldots, n \}$ be arbitrary but fixed. I am trying to prove that there ...
SebastianP's user avatar
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0 answers
40 views

Rewriting summation in binomial form - counting problem

Let $A$ be a player set and $Q$ a subset, which has been given. Furthermore, players $x,y,z \in A$. I have already managed to rewrite the following summation (where I set $n=|A|-|Q \cup \{ x,y \}|$ ...
hans15's user avatar
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1 answer
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Tensor product of basis and dual basis

In my textbook is stated the following expression: $$\hat{\delta}=\delta^i_j(\vec{a}_i\otimes\vec{a}^j)=\vec{a}_i\otimes\vec{a}^i$$ And I just can't understand what the logic here is. First of all ...
Krum Kutsarov's user avatar
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2 answers
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Counting discrete space, discrete time random walks that are always strictly positive.

Let $ \Theta \ge 1 $ and $n \ge 1 $ be integers. Consider integer sequences of length $n$ composed of entries each one running independently over the range ${\mathfrak R}_\Theta:=\left\{-\Theta,-\...
Przemo's user avatar
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Kronecker Delta as a Tensor

Let $\delta^i_j$ be the Kronecker delta function, i.e. $1$ if $i=j$ and $0$ otherwise. Then, it is easy to verify that this value is a rank 2 mixed tensor of one covariant index and one contravariant ...
Chordx's user avatar
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Partition of n into k parts with at most m

I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate $$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$ My approach was ...
Qant123's user avatar
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0 answers
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Is there any method to solve the integral including Kronecker delta, not Dirac delta?

It's a weird question to me, too. Nevertheless, please refer to below formula. $V(r_1,r_2)=\langle\int\varepsilon_\nu(R_1)\varepsilon_\nu(R_2)\frac{e^{i\omega|R_1-r_1|}}{|R_1-r_1|}\frac{e^{i\omega|R_2-...
XX X's user avatar
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1 vote
0 answers
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Contaction of Kronecker Delta with another Kronecker Delta

If a space is of dimension $d$, the Wikipedia article seems to suggest that contracting the Kronecker delta with itself gives $2d(d-1)$. But this seems confusing to me, say that I have $4$ dimensions (...
Tom's user avatar
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Taking Trace with Multiple Metrics

I am evaluating some equations where one has products of multiple metrics with $4$-momenta $q$, $p$ and $k$ and getting coordinate sickness slightly. The metric is in fact Minkowski, so we can write ...
Tom's user avatar
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1 vote
2 answers
155 views

Symmetries of the multiplication of Kronecker delta and Levi-Civita symbol?

I was pondering on the symmetries that \begin{equation} \epsilon_{ijk}\delta_{\ell m} \end{equation} might have upon interchanging indices of the Kronecher delta and Levi-Civita symbol, e.g. the ...
Bjaam's user avatar
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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
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1 vote
1 answer
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Can the convolution of two complex sequences be a Kronecker delta?

Let $c$ and $d$ be two complex-valued sequences that are absolutely summable, i.e. \begin{align*} c: \mathbb{Z} \to \mathbb{C}, \, \sum_{n \in \mathbb{Z}} |c_n| < \infty. \end{align*} We define the ...
Andreas132's user avatar
0 votes
0 answers
32 views

Fourier Transform of continuous time limit mimicking Kronecker Delta

I understand that Kronecker Delta is a function that is defined for discrete arguments as follows: $$\delta_{ij} = \begin{cases} 0, & i \ne j \\ 1, & i = j \end{cases}$$ However, I want to ...
Srini's user avatar
  • 1,022
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0 answers
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Proof of the relation between $SO(3)$ generators and their Lie braket

I'm reading a book called "Physics from symmetry" by Schwichtenberg. The author talks about generators $\{J_i\}$ of the Lie algebra of $SO(3)$ and he derives the conditions that they have to ...
Luke__'s user avatar
  • 183
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0 answers
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Simplify Ricci tensor

Given a Ricci tensor $$R_{kk} = -(g_{kk})/a^2 - \delta_{ki}\delta_{ik}g_{kk}/a^2 + \delta_{jk}\delta_{ik}g_{kk}/a^2 + \delta_{ki}\delta_{ik}g_{kk}/a^2.$$ What's its correct simplified form? Here's my ...
Pedro Italo's user avatar
0 votes
1 answer
18 views

Indices without a given domain in sum notation.

I'm learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a ...
Numerical Disintegration's user avatar
7 votes
1 answer
190 views

Evaluate: $\sum\limits_{n\ge1}\sum\limits_{m\ge0}\sum\limits_{k=0}^{n-1}\frac{y^nm^k(-n)^m\delta_{k+m-n+1}}{(k+m-n+1)!\Gamma(n-k)k!n}$

Context: The cube super root ssrt$_3(x)$ series expansion yielded part of it as: $$\sum_{n=1}^\infty\frac{y^n}{n!}\sum_{m=0}^\infty\frac{(-n)^m}{m!}\sum_{k=0}^{n-1}\binom{n-1}k\left.\frac{d^kt^m}{dt^k}...
Тyma Gaidash ٠'s user avatar
0 votes
1 answer
105 views

How is this probability distribution evaluated with Kronecker Delta and summation?

I am reading through Robert Swendsen's An Introduction to Statistical Mechanics and Thermodynamics, and in section 3.4, he provided an example of the probability of the sum of two dice using Kronecker ...
Halcyon Mo's user avatar
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1 answer
68 views

Can I exchange the indices in this particular product of Kronecker delta's?

Is the following true? (We are not using Einstein summation notation here) $$\delta_{mk} \delta_{nk} = \delta_{mn} \delta_{km}$$
Tom's user avatar
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1 answer
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Integrals of complex exponentials in terms of Kronecker deltas

Suppose that I have a set of of $L$ real parameters $\{\phi_1,\phi_2,\dots,\phi_L\}$, each of them taking values in the interval $[0,2\pi]$. I was wondering if there is a closed form for the following ...
Javier Martínez 's user avatar
3 votes
2 answers
101 views

how to factor out a term in Tensor notation

I am studying classical mechanics and I was asked in my homework to calculate poisson brackets of components of angular momentum. I couldn't understand how to approach, then I looked at the solution. ...
Itay2924's user avatar
0 votes
0 answers
62 views

Evaluate the numerical value of three Kronecker delta

How to go about evaluating the numerical value of three Kronecker delta: $$\delta_{ii}\delta_{jj}\delta_{kk}$$ Tried summing at like this: $$\delta_{ii}\delta_{jj}\delta_{kk}=\delta_{ii}\delta_{ii}\...
darkhorse1123's user avatar
0 votes
1 answer
76 views

Kronecker delta in a finite sum

I'm looking at a sum of a function which involves a trinomial expansion with $$ \sum_{i+j+k = N, 0\leq\{i,j,k\}\leq N} \binom{N}{i,j,k} f(i,j,k) $$ I started by rewritting this with $$ \sum_{i,j,k = 0}...
peep's user avatar
  • 125
0 votes
0 answers
39 views

Summation in closed form

I have the following summation: \begin{equation} f(k_1,k_2) = \sum_{l_1 = 0}^{k_1}\sum_{l_2 = 0}^{k_2} {\rho}^{l_1+l_2}\delta[k_1-k_2+l_2-l_1] \end{equation} where $\rho$ is a constant and $\delta$ is ...
lord voldemort's user avatar
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0 answers
35 views

Simplify recursive equation $h(n) = \alpha h(n-1) + (1-\alpha)\delta(n)$

Can anyone tell me how to simplify this recursive equation? $$h(n) = \alpha h(n-1) + (1-\alpha)\delta(n)$$ $\delta(n)$ is Dirac delta function. I have got to this so far: $$h(n) = (1-\alpha)\delta(n) +...
Svw's user avatar
  • 1
0 votes
1 answer
36 views

Why $a_i b_m c_n\left(\delta_{m k} \delta_{n i}-\delta_{m i} \delta_{n k}\right) \mathbf{e}_k=a_n b_m c_n \mathbf{e}_m-a_m b_m c_n \mathbf{e}_n$?

I am stuck in an intermediate step. In order to evaluate the product of the Levi-Civita symbols, we use the identity $$ \epsilon_{m n j} \epsilon_{i j k}=\delta_{m k} \delta_{n i}-\delta_{m i} \...
user avatar
3 votes
1 answer
175 views

Alternate multinomial theorem for $\frac{d^n}{dx^n}\prod\limits_{k=1}^m f_k(x)$ without $\sum\limits_{k_1+\dots+k_m=n}$ nor Kronecker delta.

The generalized product rule complicates putting series coefficients into closed or hypergeometric form. There are 2 forms with Lagrange $n$th derivative notation and the multinomial $\binom n{n_1,\...
Тyma Gaidash ٠'s user avatar
3 votes
1 answer
431 views

Is the type $(1,1)$ Kronecker delta tensor, $\delta_a^{\,\,b}$ equal to the trace of the identity matrix or always $1$ when $a=b$ and zero otherwise?

I'll ask this question using very simple examples working in flat cartesian space (just $2$ spatial dimensions). I'll be using the Einstein summation convention throughout this question, but since I'm ...
Electra's user avatar
  • 324
0 votes
1 answer
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Volume integral over dyadic: Is it true that $\int_{\mathbb{R}^3} d^3 \vec{x}\; x_i x_j f(|\vec{x}|) \ \propto \ \delta_{ij}$?

Let $f(a)$ be some function of $a>0$. Is it true that $$ \int_{\mathbb{R}^3} d^3 \vec{x}\; x_i x_j f(|\vec{x}|) \ \propto \ \delta_{ij} $$ where $\delta_{ij}$ is the Kronecker delta, and where $x_i$...
QuantumEyedea's user avatar
1 vote
0 answers
29 views

Expression with Triple Sum of Kronecker Delta

I am trying to evaluate the following term. I already have asked a question to a similar problem. $$\sum_{c_1}^N\sum_{c_2\ge c_1}^N\sum_{c_3\ge c_1}^N \delta_{c_2,c_k}P_{c_1,c_2}\delta_{c_3,c_l}P_{c_1,...
eeqesri's user avatar
  • 729
1 vote
1 answer
41 views

Expression with Kronecker Delta over upper triangular Matrix

I am having trouble evaluating the following expression involving Kronecker delta's over a double sum restricted over the upper triangular terms: $$\sum_{c_1=0}^N\sum_{c_2\ge c_1}^N \delta_{c_2,l}\...
eeqesri's user avatar
  • 729
0 votes
0 answers
30 views

Tensor calculus: problem with second order delta systems property

The properties described in this question come from P. Grinfeld's book: "Introduction to Tensor Analysis and the Calculus of Moving Surfaces". The Kronecker delta (or symbol) is defined by ...
Meclassic's user avatar
  • 425
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0 answers
119 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
48 views

Sum Simplification

For $q,k\in\mathbb{N}$ and $1\leq q\leq N$, is the following simplification $$ f(q,k)=\sum_{j=1}^Ne^{-(2k\pi\text{i})jq/N}=N\sum_{i=1}^k\delta_{iq,N} $$ correct? Here, $\delta_{i,j}$ is the Kronecker ...
sam wolfe's user avatar
  • 3,417
2 votes
1 answer
166 views

Matrix written in terms of Kronecker delta

What would be the matrix form of $P$ defined in equation (1.11) of this published work, where (I have chosen $n=2$, for simplicity) $$P_{kl}=\begin{cases} \delta_{k,2l-1} ~~~ k \le 2\\\\ \delta_{2+...
User101's user avatar
  • 504
1 vote
1 answer
53 views

Is there an expression for $i, j, k \in \left\{1,\, 2,\, 3\right\}$ with $i \neq j \neq k$?

I want to make a statement where $i, j, k \in \left\{1,2,3\right\}$ but $i \neq j \neq k$ and assumed there'd be an equivalent of $$ \delta_{ij} = \begin{cases} 0, & i = j \\ 1, & i \neq j \...
Rax Adaam's user avatar
  • 1,186
0 votes
1 answer
503 views

Levi-Civita and and Kronecker delta identity

I'm wondering how $\epsilon_{ijk}\epsilon_{ilm} $ can give this matrix $\begin{pmatrix} \delta_{ii} & \delta_{ij} & \delta_{ik} \\ \delta_{li} & \delta_{lj} & \delta_{lk} \\ \delta_{mi}...
Redwaves's user avatar
0 votes
3 answers
4k views

Why does $\epsilon_{ijk}\epsilon_{ijk} = 6$?

This is how I began, Proof. Using $$ \epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \tag{1}\label{eq1} $$ I wrote it as $ \epsilon_{ijk}\epsilon_{ijk} = \delta_{jj}\...
Micheal S. Bingham's user avatar
6 votes
1 answer
1k views

Is this representation of the Kronecker delta as a summation correct?

Wikipedia provides the following representation of the Kronecker delta: \begin{equation}\label{eq1} \delta_{jk} = \frac{1}{N}\sum_{n=1}^N \mathrm{e}^{\mathrm{i}2\pi(j-k)n/N} \end{equation} At first ...
Involute's user avatar
  • 273
1 vote
0 answers
360 views

Dirac Delta vs Kronecker Delta for Discrete Fourier Transform

As I understand it, the Dirac Delta function is commonly associated with the Fourier Transform because it produces exponential functions that can represent signals. However, as the Dirac Delta ...
Hector Lombard's user avatar
-1 votes
1 answer
60 views

Summation with Kronecker delta [closed]

I am having some trouble in understading why $$\sum_{m=0}^{\infty} e^{mix}\delta_{k+m} = e^{-ikx}$$ which was the result i got from wolfram
curryxd's user avatar
  • 33
0 votes
1 answer
683 views

Evaluate this Kronecker Delta multiplication

I need to evaluate this multiplication of kronecker deltas $$\delta_{ij}\delta_{km}\delta_{jk}\delta_{im}$$ seems to be a very simple exercise, but my question is: Can I change the order of the deltas ...
Chris Schwenke's user avatar
1 vote
1 answer
254 views

What should be the output of a double sumation over Kronecker delta?

$$\left(\sum_{i=1}^3 \sum_{j=1}^3 \delta_{ij}\right)\left(\sum_{m=1}^3 \sum_{n=1}^3 \delta_{mn}\right)$$ eq. 1. Should be: I am just confused because I am getting $3 \times 3$. I was reading a book. ...
Rafael Acosta's user avatar