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Questions tagged [kronecker-symbol]

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

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what is squared of a Kronecker ij?

Is it right to write $\delta_{ij}\delta_{ij}=(\delta_{ij})^2=\delta_{ij}$?
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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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135 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 ...
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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 $|...
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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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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: $$...
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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 ...
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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 ...
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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 ...
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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_{...
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Proof of $\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

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}...
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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}-\...
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557 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, ...
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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 = -\...
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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{...
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112 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_{...
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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'...
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1answer
382 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 ...
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1answer
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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 ...
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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 ...
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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}...
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1answer
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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_{...
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1answer
545 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 ...
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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 ...
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1answer
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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 ...
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1answer
633 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^{...
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1answer
131 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 ...
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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 $$\...
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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 ...
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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\...
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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{...
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1answer
366 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 $\...
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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 $T_{ij}$ ...
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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 $\lfloor \rfloor$ ...
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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 : ...
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Show $\int v^2 \left( v_i v_j - \frac13 v^2 \delta_{ij} \right) f_m d^3v = 0 ,$

I need to show, that $$\int v^2 \left( v_i v_j - v^2 \delta_{ij} \right) f_m d^3v = 0 ,$$ where $$f_m \propto \exp(-v^2), $$ is Maxwellian distributin. Actually, those indicies frustrates me, I know ...
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Levi Civita product proof

Does anyone know/know where to find the proof of: $$\epsilon_{ijk}\epsilon_{lmn} = +δ_{il}δ_{jm}δ_{kn} + δ_{im}δ_{jn}δ_{kl} + δ_{in}δ_{jl}δ_{km} −δ_{im}δ_{jl}δ_{kn} − δ_{il}δ_{jn}δ_{km} − δ_{in}δ_{...
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Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
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Application of Legendre, Jacobi and Kronecker Symbols

Legendre, Jacobi and Kronecker Symbols are powerful multiplicative functions in ...
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Why does the Kronecker symbol $\left(\frac{a}{\cdot}\right)$ define a character?

Let $a\not\equiv 3\pmod 4$ and $a\ne 0$. How do you show that $\chi(n):=\left(\frac{a}{n}\right)$ (where $\left(\frac{\cdot}{\cdot}\right)$ denotes the Kronecker symbol) defines a character of modulus ...
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Sum of a product of four Kronecker Deltas

The Kronecker delta has the following property: $$\sum_{k} \delta_{ik}\delta_{kj} = \delta_{ij}. $$ Does anyone know whether the following formula is correct? $$\sum_{i=1}^N \delta_{ij}\delta_{ik}...
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Kronecker Zeta function

If we define the Kronecker symbol K(a,n) as at Wikipedia, can we define $$\zeta_K(s,a) = \sum_n \dfrac{K(a,n)}{n^s}$$? If so, what does it equal?