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 ...
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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 ...
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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 :
$$\delta_{ij}=\cases{1&if $i=j$\\...
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Two similar proofs that $\frac{\partial}{\partial x'^\mu}={\Lambda_\mu}^\nu\frac{\partial}{\partial x^\nu}$, which one is correct?
Using the chain rule, show that the derivative transforms as
$$\frac{\partial}{\partial x^\mu}\to\frac{\partial}{\partial x'^\mu}={\Lambda_\mu}^\nu\frac{\partial}{\partial x^\nu}\tag{A}$$
This is the ...
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Sum with multiple Kronecker deltas
I have a problem dealing with the following expression:
$$\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} $$
I ...
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Why does the summation symbol disappear?
We know the orthogonality condition
$$\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}
$$
From earlier in the text we ...
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Find a matrix such that the quadratic form of a orthonormal basis is equal to the Kronecker delta
For $n \in \mathbb{N}$ let $\{v_1, v_2, \ldots v_n \}$ be a orthonormal basis of $\mathbb{R}^{n}$. Further, let $i \in \{ 1, 2, \ldots, n \}$ be arbitrary but fixed. I am trying to prove that there ...
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Rewriting summation in binomial form - counting problem
Let $A$ be a player set and $Q$ a subset, which has been given. Furthermore, players $x,y,z \in A$. I have already managed to rewrite the following summation (where I set $n=|A|-|Q \cup \{ x,y \}|$ ...
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Tensor product of basis and dual basis
In my textbook is stated the following expression:
$$\hat{\delta}=\delta^i_j(\vec{a}_i\otimes\vec{a}^j)=\vec{a}_i\otimes\vec{a}^i$$
And I just can't understand what the logic here is. First of all ...
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Counting discrete space, discrete time random walks that are always strictly positive.
Let $ \Theta \ge 1 $ and $n \ge 1 $ be integers. Consider integer sequences of length $n$ composed of entries each one running independently over the range ${\mathfrak R}_\Theta:=\left\{-\Theta,-\...
- 11.5k
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Kronecker Delta as a Tensor
Let $\delta^i_j$ be the Kronecker delta function, i.e. $1$ if $i=j$ and $0$ otherwise. Then, it is easy to verify that this value is a rank 2 mixed tensor of one covariant index and one contravariant ...
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Partition of n into k parts with at most m
I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate
$$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$
My approach was ...
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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-...
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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 (...
- 3,005
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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 ...
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