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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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Kronecker delta from cardinal sine

One possible definition for Dirac's delta function is via a limit of the cardinal sine, according to \begin{equation} \lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a} \mathrm{sinc}\left(\frac{x}...
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Matrix equations for derivative schemes using Taylors Theorem

In a tutorial, I have been given an expression for the $k^{th}$ derivative (which is a combination of an expression for the $k^{th}$ derivative and Taylors Theorem): $$ \frac{d^{p}}{dx^{p}}f(x) \ = \...
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What does the symbol $\delta$ mean on this page?

On the Wikipedia page on Arithmetic Functions, the section Relations Among The Functions makes frequent references to a variable $\delta$ (or is it a function? Some other kind of value?). It's ...
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finding the moments of a spline defined by Kronecker delta

having the Spline function of equidistant knots $x_i$ defined by $S_j(x_k)=\delta_{jk},$ where $j,k=0,1,2,...,n$ and $S_j^{(2)}(x_0)=S_j^{(2)}(x_n)=0$ , how can I find the moments $M_1,...,M_{n-1}$ ? ...
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Ambiguity with Kronecker Delta function

Given $Q = \sum_{i j s r} [(a_{i j s} − \frac{k_{i s} k_{j s}}{2m_{s}}) \delta(s,r) + c_{jsr} \delta(i,j)] \delta(\gamma_{i,s},\gamma_{j, j})$ Where $\delta$ is Kronecker function. I am having ...
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Identify all summation indices,

For the following expression, $(i)$ Identify all summation indices, $(ii)$ simplify the expression to such a form so that no Kronecker deltas appear in the expression. $$(a) \ A_{\alpha} B_{\beta} ...
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A question on deriving a kronecker delta identity

I've recently come across the following identity: $$\frac{1}{\sqrt{n!m!}}\bigg(\frac{\mathrm{d}}{\mathrm{d}Z^{\ast}}\bigg)^{m}\big(Z^{\ast}\big)^{n}\bigg\vert_{Z^{\ast}\to 0}=\delta_{n,m}\;.$$ Here is ...
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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, ...
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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{\...
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Kronecker delta for inequality

Kronecker delta return 1, or 0 depending on a conditional statement (if $i = j$), for example, $\delta_{i,j} = 1$ if $i = j$, and $\delta_{i,j} = 0$ otherwise. I would like to know if there are ...
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Relation between the differential operator, a vector and the Kronecker delta

I saw in a textbook, $D_{c}x_{i} = \delta_{ci}$. Could someone explain this to me?
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Understanding of the Kronecker delta

From my understanding $e_a\cdotp e_b$ = $\delta_{ab}$ (for $a,b = 1,2,3$) equals $1$ when $a=b$ and $0$ when $a \neq b $ where $e_a$ and $e_b$ are vectors with entries 1 in the $a$'th and $b$'th row, ...
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Levi-Civita and Kronecker delta notation

I was wondering how to do the following $\epsilon_{ijk}\sigma_{jk}=\epsilon_{iji}=0$ I get this is $0$ but don't understand how they got the $\epsilon_{iji}$
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what does it mean to be symmetric for tensors and Kronecker delta symbols and help explain this answer to me

i understand how to change 2 tensors into Kronecker delta symbols but unsure how they managed to transform back to just one. If someone could add all the steps to get to the answer that would be ...
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Evaluating $\int_{-1}^{1} x P_{n}(x) P_{m}(x) dx$

I was trying to evaluate - $\int_{-1}^{1} x P_{n}(x) P_{m}(x) dx$ Point is there is a $x$ in the integrand, otherwise, it is a well known integral which results to $\frac{2}{2n+1}$ if $m=n$ and $0$ ...
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Where did the formula relating a contravariant and covariant basis come from?

Given coordinate axes that are not necessarily orthogonal with independent basis (and not necessarily unit) vectors $\newcommand{\b}[1]{\hat{\bf #1}} \b e_1$ and $\b e_2$, a vector $\newcommand{\v}[1]{...
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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 ...
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Evaluating Expressions with Kronecker Deltas

I am trying to learn about vector/tensor math and stumbled across these exercises in my textbook. (1) Evaluate the expression $\delta _j^i \delta _i^j $ (2) Evaluate the expression $\delta _i^i \...
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Einstein's summation convention with multiple terms

Let $$N_{ij} = \delta_{ij} - \epsilon_{ijk}n_k + n_in_j$$ and $$M_{ij}= \delta_{ij} + \epsilon_{ijk}n_k$$ where $n_i$ is the cartesian coordinates of the unit vector $\textbf{n}$. Show that $$N_{ij}...
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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, ...
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Does changing the order of the indices of the Kronecker delta within a summation matter?

In my notes I have that $$\sum_m a_m \delta_{nm}=a_1\delta_{n1}+a_2\delta_{n2}+a_3\delta_{n3}+\cdots=a_n\tag{A}$$ Is this really correct? I thought that for the Kronecker delta the first index ...
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Formula for Cardinality of Finite Sets [closed]

I recently found this formula. How do I prove it? Let $S$ be a finite set which contains no repeats (every element is unique). Then $$|S|=\sum_{x\in S}\delta_{xx}$$ Where $\delta_{ij}$ is the ...
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Quantum Mechanics - Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
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Do these integrals evaluate to terms involving Kronecker Deltas?

There is a famous integral representation of the Kronecker delta: $$ \delta_{N,M} = \int_0^1 dx\ e^{-2 \pi i (N-M)x} $$ Noting this, I have encountered two integrals where $N,M,m \in \mathbb{Z}$. ...
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Three index entities summing to product of Kronecker deltas

While looking at this problem I ended up having to find entities $a_{ijp}$ and $b_{klp},$ with $i,j,k,l,p = 1, \ldots, n,$ such that $$\sum_{p=1}^{n} a_{ijp} \, b_{klp} = \delta_{ik} \, \delta_{jl},$$...
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Equation containing cofactor of derivative and Kronecker-delta

Let $d_{ij}$ be the cofactor of $\frac{\partial f_j(x)}{\partial x_i}$ in $J_f(x)$, i.e. $d_{ij}$ is $(-1)^{i+j}$ times the determinant which you obtain from $J_f(x)$ cancelling the $j$-th row and the ...
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Delta/metric question (context commutator poincare transf.)

The problem statement, all variables and given/known data Relevant equations I believe that $\frac{\partial x^u}{\partial x^p} =\delta ^u_p $ (1) $\implies $ (if $\delta^a_b $ is a tensor, I'm not ...
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The Kronecker delta symbol and indicial/Einstein notation. (Help solving problem in introduction to tensor calculus.)

Problem Statement I have attached an image, below, which shows an exercise in the book "An Introduction to Tensor Calculus" by Jacques L. Mercier. I have been trying to solve the exercise shown (...
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Trouble with delta kronecker

I am having trouble understanding when it is correct to use the kronecker delta function. Let me elaborate. I am given a symmetric bilinear form with its corresponding equation (that tells us what ...
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How is the dot product of two cartesian unit vectors equal to the Kroencker delta?

In my lectures, the relationship $\underline{\hat{e_k}} \cdot \underline{\hat{ e_j}}$ is given to be equal to $\delta_{kj}$ - how is this so? From my understanding this is equal to $ \left( \begin{...
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How to proof that a dual basis applied to the basis give the Kronecker delta?

Since a long time I was wondering how can we build proof that a basis applies to its dual give the Kronecker delta. I'm now following on-line lectures related to topology and find out that there is a ...
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Looking for ${f_n}$ such that $\int_0^1 (x-t)^{m-1}f_n(t) dt = \delta_{n,m}$

Good day, I am wondering whether it is possible to find a sequence of functions $f_n$ such that $$\int_0^1 (1-t)^{m-1}f_n(t) dt = \delta_{n,m}$$ for every $0<n,m$. Thank you.
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Simplifying Product of Kronecker Delta Functions

Im attempting to multiply the infinite matrices $$A = \begin{bmatrix} 0 & \sqrt 1 & 0 & 0 & 0 & ... \\ \sqrt{1} & 0 & \sqrt{2} & 0 & 0 & ... \\ 0 & \sqrt 2 &...
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Kronecker Delta and Constants

I have a small question and I hope I make this clear. Would one distribute the Kronecker Delta as follows? $\sum_{j}^n\ ( C + x_j ) \delta_{ij}$ $= \sum_{j}^n\ ( C\delta_{ij} + x_j\delta_{...
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Kronecker delta question!

I am trying to grasp the basics of tensor calculus. I've come across the Kronecker delta and it confused me. $$\delta_j^i=\left\{\begin{aligned} &1, i=j\\ &0,i\not=j \end{aligned} \right.$$...
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Relationship between summation and convolution

Currently I am reading a paper which derives a multiplication of matrix such as: $AB=I$, with $B$ is the inverse of $A$, $I$ is an identity matrix. This is the short hand notation for the following ...
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Proving that a set of functions with the Dirichlet convolution operation is a group

I want to prove that the set $S:(f:\Bbb Z\to\Bbb R);f(1)\neq 0)$ with the Dirichlet Convolution operator is a group. My process: $f,g:\Bbb Z\to\Bbb R$ and $f*g:\Bbb Z\to\Bbb R$ so $f*g\in S$ $f*(g*h)...
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Multiplying shifts of Kronecker Delta

Can someone please explain to me these two equations? $$\delta_{n}\delta_{n-2k}=\delta_{n}$$ and why $$\delta_{n-1}\delta_{n-2k}=0$$ and why $$\delta_{n}\delta_{n-2k-1}=0$$ Any help would be ...
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Definition of Elementary Matrices

I'm a bit confused about the definition of elementary matrices which are used to represent elementary row operations on an extended coefficient matrix when doing the Gaussian elimination. In my ...
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Proof of $(\mu*1)=\delta_1$ where $(f*g)$ is Dirichlet Convolution

I was interested in the proof for this fact because it is used to prove M$\ddot o$bius Inversion Formula. However, I did not completely understand how proof wiki used a series of binomial coefficients ...
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Einstein Summation Notation and Kronecker Delta Problem

Evaluate $\delta_j^iv_iu^j, \delta^2_j\delta^j_kv^k,$ and $\delta^3_j\delta^j_1$. The Kronecker-delta: https://en.wikipedia.org/wiki/Kronecker_delta Einstein notation: https://en.wikipedia.org/wiki/...