Questions tagged [index-notation]

For questions about index notations, e.g. abstract index notation, Einstein summation convention, topics in introductive tensor calculus, Levi-Civita Symbol, Kronecker Delta symbol, proofs of vector calculus identities or fluid dynamics formulae using index notation.

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22 views

Problem in the derivative with multi-index notation

I'm currently having some problems in understanding what our professor wrote. Text reads: $$\partial^{\alpha}\left((x-x_0)^{\alpha} \phi\left(\frac{x-x_0}{\epsilon}\right)\right) = \alpha!\phi\left(\...
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49 views

Recursive indexation problem

Consider a hexagonal lattice indexed with relative indexation $(q,p)$, for a fixed cell, as follows Now consider $S_k$ as the set of relative indexes of the $k$th ring of cells around the focused ...
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1answer
59 views

How do you call this system?

Is there a specific name for a dynamical system that depends on the relative indexation $i\pm k$ for some $k$? For example, consider the following dynamical system defined on a ring of cells by $$ \...
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1answer
21 views

Index notation in question about distribuitivity of tensor product over direct sum

Thanks to @peek-a-boo I edited my question In this post Tensor product and direct sum the author has the map $\varphi : (\bigoplus_\alpha M_\alpha)\times N\to \bigoplus_\alpha (M_\alpha\otimes N)$ ...
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1answer
81 views

How to write an "index set"

Imagine I have a finite sequence $\{s_i\}_{1\leq i\leq N}$, for some $N$. Now assume I want to sum all the terms of such sequence apart from the term $s_i$, for some fixed $i$. One way to do it is to ...
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2answers
159 views

Large Vector Notation

Imagine I have an arbitrary vector $\mathbf{v}\in\mathbb{R}^n$ and say it can be represented as $$ \mathbf{v}=(v_1,...,v_n)^T $$ where it's understood that the indexes are in increasing order. Imagine ...
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5answers
159 views

What is the correct way to denote empty n-ary intersection of sets?

I'm trying to prove the statement: Show that a finite intersection of open subsets in a metric space is open. If I'm able to enumerate the sets as $U_1, \ldots, U_n$ and consider $U := \bigcap\...
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1answer
21 views

The set of all the summands/factors in a sum/product

An example: The sum $\displaystyle = \sum_{k=1}^n k$ Therefore, the set $= \{1,...,n\}$. In this particular case, one could of course just use $\Bbb N^n$, but in other cases, this would not be the ...
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29 views

Sum of 4 first index $\sum_{n=1}^4a_i=1$

$\sum_{n=1}^4a_i=1$ The meaning of this sum, means that: *$a_1=1, a_2=1, a_3=1, a_4=1$ OR: *$a_1=1, a_2=2, a_3=3, a_4=6$ Or something else?
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2answers
88 views

Proof for the Laplacian of scalar fields using index notation [closed]

For scalar fields Φ and Ψ, the Laplacian is defined by $$∇^2(ΦΨ)=(∇^2Φ)Ψ+2∇Φ\cdot∇Ψ+Φ∇^2Ψ$$ where $∇$ is the usual del operator and $∇^2$ is the Laplacian. How can I prove this relation? I tried the ...
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64 views

Is the author messing up the notation or am I thick?

In Arfken and Webber's "Mathematical Methods for Physicists" under the topic "General Tensors" the following is given: $$dx^i=\frac{\partial x^i}{\partial q^j}dq^j$$ or in vector ...
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1answer
77 views

How can I prove that the differential operator with respect to covariant coordinates behaves contravariantly?

Given the Operators $\partial_i=\frac{\partial}{\partial x^i}$ and $\partial^i=\frac{\partial}{\partial x_i}$ I am supposed to show that they behave co- and contravariantly (as implied by the ...
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1answer
38 views

How to deal with products of “shifted” Kronecker deltas?

How does one deal with expressions like $$ \sum_\mu s_\mu \delta_{2 \mu-1,j} \delta_{2\mu,k} \quad \text{ or } \quad \sum_\mu a_\mu\delta_{2\mu,j} \delta_{2\mu -1,k} \quad ? $$ The usual rules would ...
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1answer
56 views

What is a varible produced through binary dimension reduction called?

If I take k binary variables and reduce them to one k-long string variable consisting of zeros and ones. What would you call the new variable? For example, for unit i, xi1 = 1, xi2 = 0, xi3 = 0, and ...
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1answer
47 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 \...
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18 views

Notation for raising a number to elements of a vector.

If I had a number $n \in \mathbb{C}$, a $d$-length vector $\mathbb{x}=\{x_1,x_2,\cdots,x_d\}$ and another $d$-length vector $\mathbb{y}=\{y_1,y_2,\cdots,y_d\}$, where $y_i=n^{x_i}$ for $i=1,2,\cdots,d$...
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1answer
52 views

Einstein notation interpretation

I have the following function: $$ \alpha = \alpha_0 + \frac{1}{2} V_{ij}V_{ij} + \frac{1}{3} V_{ij}V_{jk}V_{ki}+\frac{1}{8}V_{ij}V_{ij}V_{kl}V_{kl} $$ which is a scalar and depends on the $V$ matrix: $...
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1answer
48 views

Prove $T\Omega \times T x=T(\Omega \times x)$

Let $T:V\to V, T\in SO(3)$ represent a rotation matrix in $\mathbb R^3$, $\Omega \in \mathbb R^3$ is a vector, and $x\in \mathbb R^3$ also represent a vector. How to prove $T\Omega \times T x=T(\Omega ...
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1answer
28 views

Formula for the Maxwell Stress tensor in arbitrary coordinates

This question is nearly identical to my last, except this time its the Maxwell stress tensor, not the Cauchy stress tensor. I often see its components written as $$\sigma_{ij}=\varepsilon_0E_iE_j+\...
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0answers
15 views

A question about multi-indices notation

I'm reading a paper which introduces the following notation. Let $[n]:= \{1, 2, \dots n\}$. Then consider $\textbf{a,b} \in [n]^l$. I guess that they are referring to so-called multi-indices that I'm ...
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1answer
63 views

Is there a consistent formula for the Cauchy stress tensor in fluid mechanics?

Recently, my undergraduate thesis advisor gave me a formula for the Cauchy stress tensor in fluid mechanics: $$\boldsymbol{\sigma}=-P\mathbf{I}+\mu\big(\nabla\boldsymbol u+(\nabla\boldsymbol u)^{\...
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1answer
46 views

Rules for using multi-indices [closed]

I've only encountered multi-index notation in the context of the multinomial theorem. There, the notation is used like this: $(k_1 + k_2 +...+k_m) = n, \ k_i \in \Bbb N_{\ge 0}^n$, iterate over every ...
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1answer
40 views

Issues understanding the multinomial theorem and its multiindex notation

$$(x_1+x_2+...+x_m)^n=\sum_{(k_1 + k_2 +... +k_m) \ = \ n} {n \choose k_1,k_2...k_m} \prod^m_{t=1}x_t^{k_t}$$ Let's do $(a+b+c)^3$. That means $a =x_1, b =x_2, c=x_3=x_m$. The multiindex below the ...
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25 views

Repeated indices on the same variable in Einstein notation

I've been looking at this paper: "Marginal Curvatures and Their Usefulness in the Analysis of Nonlinear Regression Models". I don't think this paper is freely available. I'm interested in ...
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1answer
45 views

Levi-Civita symbol with matrix

What is the matrix form of $\epsilon_{ijk} A_{jk}$?
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1answer
99 views

About Tensor Index notation

In learning about (Einstein) tensor index notation I've noticed what I feel are inconsistensies with the notation. Now in Einstein notation vectors are denoted by superscripts $v^i$ while co-vectors ...
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1answer
87 views

What is the difference between these sum notations? $\sum_i \sum_j a_i b_j \overset{?}{=} \sum_{i,j} a_i b_j$

I saw both written in a textbook (cannot remember where exactly), but I do not know what the difference should be between these: $$\sum_i \sum_j a_i b_j \overset{?}{=} \sum_{i,j} a_i b_j$$
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38 views

Upper and lower case indices in matrix notation

In my matrix algebra book, Matrix Algebra Useful For Statistics by S. Searle and A. Khuri, I came across several equations that include upper and lower case indices for rows and columns. This ...
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66 views

Partial derivatives using Einstein / Index notation

I've been trying to compute a partial derivative of an arbitrary function for a while using index notation but I think I am missing something. Let $\zeta$ be some well behaved scalar function which we ...
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4answers
111 views

Why does $15\sqrt{15} = 15^{3/2}$?

I thought I understood the process of converting a surd to index form, but for the challenge: $15\sqrt{15} $, I don’t understand why $15^{3/2}$ would be the answer (according to the book I’m working ...
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35 views

How do you compute a weak formulation for the Hooke's Law using Einstein notation?

I am interested in producing a weak formulation for linear elasticity for a finite element scheme but am stuck when checking my indices with Einstein notation. I am looking at the following PDE: $$ \...
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2answers
85 views

How is $l^2(\mathbb{R})$ defined?

Across many texts, I have seen that $l^2$ can be defined over $\mathbb{N}$ (denoted $l^2(\mathbb{N}$) or $\mathbb{R}$ (denoted $l^2(\mathbb{R}$) or orther spaces. I am not sure what does this notation ...
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2answers
81 views

Indexing a set of objects with a symmetry or antisymmetry property

Suppose I have a collection of objects $\{X(i, j)\}_{i,j=1}^{n}$ (which could be anything - numbers, functions, matrices, $\ldots$) indexed by $1 \leq i,j \leq n$, such that the following holds: $$ X(...
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1answer
50 views

What do these tensor partial derivatives mean?

In the Wikipedia page on Ricci calculus the following tensor derivative equation is given: $$A_{\alpha \beta ..., \gamma}:= \frac \partial {\partial x^\gamma}A_{\alpha \beta ...}.$$ However, what does ...
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68 views

determinant in tensor notation

I'm reading Pavel Grinfeld's book "Introduction to tensor analysis and the calculus of moving surfaces". I've reached the chapter where the author talks about determinants; he starts using ...
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1answer
51 views

How to write out a percentile rank thresholding for a vector as a mathematical notation?

I'm attempting to write out an operation as a mathematical formula. I've been able to code it, but I'm stuck when writing it as a concise formula. Essentially, I'm trying to extract a number of ...
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0answers
16 views

Indexing The Nodes of a Dendrogram

Suppose I have a dataset $\{x_i\}_{i=1}^n$ that I have implemented some hierarchical clustering algorithm (e.g., Single-Linkage Clustering) on. This gives me a dendrogram, which we can use to ...
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0answers
37 views

What is the correct notation for the sequence spaces $l_p$? Is it written with the $p$? in subscript, or superscript Are both versions correct?

I have a little question concerning the notation of $l^p$ spaces - the spaces where elements are p-power summable infinite sequences. In literature, I have seen the notation with $p$ as either upper, ...
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1answer
32 views

Given a NxN grid and an integer calculate the row and column [closed]

I know this is most likely elementary school math, but I am having hard time finding a good algorithm. Given a NxN grid and a number which represents the nth tile counting left to right like a ...
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76 views

Which one is more correct?$V_y$ or $V_i$

In the theorem $26.3 $: every compact subspace of a hausdorff space is closed Munkre say that the collection $\{V_y|y \in Y\}$ is a covering of $Y$ by sets open in $X$ But many authors write ...
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41 views

Am I interpreting this notation correctly?

I'm trying to parse a paper about a new (to me) clustering algorithm into code, and I want to make sure I understand the math correctly... but the notation is holding me up. From page 3 , equation 2: $...
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0answers
104 views

Product rule identity for $\nabla \times (\mathbf{A} \cdot \mathbf{u})$, the curl of a tensor field times a vector field

I am looking to derive product rules for the curl of a 2nd-order tensor field contracted with a vector field (matrix vector multiplication), $$\nabla \times (\mathbf{A}(\mathbf{x}) \cdot \mathbf{u}(\...
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0answers
16 views

Help me understand how FFT computes a particular sum with index shifting

I am interested in computing a particular sum, namely: $\sum_{j=l}^{m}(i-j)f_{i-j}g_j$. I have two polynomials defined $f(x)$ and $g(x)$. The idea is to compute the sum above by writing two new ...
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1answer
37 views

Changing the index of the given sum

Calculate $f(i)$ from the following equation of sums: $$\sum_{i=n^4}^{n^6}a_{(i+52)^7}=\sum_{i=n^3-70}^{n^9-70}a_{f(i)}$$ Using 1st and last limits of the sum we have, $f(n^3-70)=(n^4+52)^7$ and $f(n^...
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49 views

Double Summation index change

The generating function of the Bessel function can be written as: $$g_{(x,t)} =\sum_{r=0}^{\infty}\sum_{s=0}^{\infty} (-1)^s (\frac{x} {2})^{r+s} \frac{t^{r-s}} {r!s!} $$ Changing the summation index $...
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0answers
65 views

Does It Makes Sense to Have the Index of a Sum in the Argument of a Function? Two Examples Given with Integral and Sum.

Question 1: I have recently tried out answering a couple of questions with the method of using the same variable in the argument of a function and a sum for example: $$\int_0^N (\sqrt{x+1}-\sqrt x)^n ...
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1answer
61 views

Confusion regarding the use of horizontal padding of $(1,1)$-tensor indices: which is the "correct" interpretation?

Fix finite-dimensional vector spaces $V,W$. My whole life I'm been content with viewing linear maps $T:V\to W$ as $(1,1)$-tensors, i.e. elements of $W\otimes V^*$: such a $T$ yields a bilinear map $B:...
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1answer
229 views

Related expressions & non integral forms of $-2π\sum_{k=1}^∞\sum_{k=1}^n \binom{n-1}{k-1}\frac{(-1)^{k+n}+(-1)^n}{k^n}=1.774473…$ in special f(x) etc.

I have figured, hopefully, out the volume of the figure of $$(y-x^x)(y-x^{-x})=0\le x\le 1$$ about the line y=1. This volume came out to be $$V=2π\int_0^1 (1-x)(x^{-x}-x^x)dx= \boxed{-2π\sum_{n=1}^∞\...
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1answer
33 views

Notation $\sum_{k \neq k'}$: is the order important?

I am reading The Elements of Statistical Learning. At pag. 309 the Gini index is defined as: $$ \text{Gini} = \sum_{k \neq k'} \hat{p}_{mk} \hat{p}_{mk'} $$ where the index $k = {1,2,\dots,K}$. In ...
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1answer
25 views

$f(a+h)= \sum_{k=0}^{n} (1/k!)f^{(k)}(a) h^k$ for any polynomial f(x) (Comparing summations)

My final goal is to show that for any polynomial $f(x)∈F[x]$ , $f(a+h)= \sum_{k=0}^{n} (1/k!)f^{(k)}(a) h^k$ (where $F$:field and $a, h∈F$) I expanded $LHS$ and $RHS$ terms and arrived to the point ...

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