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Questions tagged [index-notation]

Use for problems relating: 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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What is a tensor with two upper or lower indices?

I know very little about tensors. I am trying understand if the following expression would make $$x^a=L^a_{\;b} M^{bc}\hat{x}_c.$$ Einstein summation convention applies. So here is what I think I ...
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How does index notation work in hermitian spaces?

So, I know in orthogonal spaces (real vector spaces with a symmetric bilinear form) there is a canonical isomorphism bewtween $E$ and $E^*$ induced by the bilinear form $\langle\vec{v}|\vec{w}\rangle=...
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2answers
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Matrix and two vectors product

Does anyone know how to represent the product of an $n×n$ matrix and two $n$-vectors in a compact form using $\sum$ and indexes? If it was only a vector then $\sum_{j=1}^{n} A_{i,j}v_j$. But with two ...
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Zero Mean Normalized Cross Correlation in Einstein Notation

I try to formulate the Zero-Mean Normalized Cross-Correlation in Einstein Notation. Thus without the Sigma's and with indices. I came up with the following but im not sure if the is correct and ...
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4answers
61 views

What is $10^{40}$ as a number? [closed]

What is $10^{40}$? Every time I google this question I get $1\mathrm{e}\!+\!40$ but I don’t understand this so what is it as a number?
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2answers
35 views

Derivatives Across Summations

So, I took one intro course in Tensor calculus and this problem reminds of that, except I can't quite recall how derivatives work with respect to components, or what those derivatives produce. ...
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1answer
32 views

Strange sum notation

I have the following sum and I would like to generalize the notation but I don't see how to choose i index... $$ -2P(A_{1} \cap A_{2}) - 2P(A_{1} \cap A_{3}) - 2P(A_{2} \cap A_{3}) = -2\sum_{i=...
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0answers
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Examples of Tensor Transformation Law

Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is ...
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1answer
24 views

Confusion regarding summation convention.

The question goes as follows: If $\theta$ is the angle between two non-null vectors $A^i$ and $B^i$, show that $sin^2 \theta = \frac{(g_{ij}g_{kl}-g_{ik}g_{jl})A^iB^kA^jB^l}{(g_{ij}A^iA^j)(g_{kl}B^...
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0answers
11 views

Proving that the covariant derivative of a vector-valued tensor is a tensor

I'm working through Pavel Grinfeld's Introduction to Tensor Analysis and the Calculus of Moving Surfaces and I'm stuck on Exercise 133: For a contravariant vector $\mathbf T^i$, prove that $\...
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2answers
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Confusion regarding summation convention

In tensor calculus, I recently came across the formula for the angle between two vectors (non null) in Riemannian Space, which is as follows: $ cos \theta = \frac{g_{ij}A^iB^j}{\sqrt {g_{ij}A^iA^j}\...
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1answer
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A (simple?) matter of notation

I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $i\in\mathbb{N}$, denoted in the sequel as $\langle a_n\rangle$ ...
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What does a dot mean in matrix element index?

Seen here https://stat.ethz.ch/education/semesters/ss2012/ams/slides/v4.1.pdf (14th slide), the formula to compute bij, we have, for instance, the elements di. (notice the dot in place of j) and d.. ...
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3answers
42 views

Matrix notation $i$ $j$

Let $A = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \\ \end{bmatrix}$ be a $n \times n$ matrix such that $a_i \cdot a_i = 1$ for all $i$ and $a_i \cdot a_j = 0$ for all $i \neq ...
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2answers
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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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0answers
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Show the cross product of the divergence of a dyad in index notation.

r is a vector given by $\sum_{i=1}^3{\mathbf{\delta}_ix_i}=x_i$, and its magnitude is $\sqrt{x_ix_i}$. v is a vector valued function of r. Show using index notation that $\mathbf{r}\times[\nabla\...
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1answer
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Relative eigenvalues and the Rayleigh quotient in tensor notation

I'm working through Pavel Grinfeld's Introduction to Tensor Analysis and the Calculus of Moving Surfaces and I'm very stuck on exercise 118, which reads: Show that the eigenvalues of the ...
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0answers
32 views

How define formally a user-defined binary operation?

Let $x \in \{0,1\}^{|\mathcal{S}|}$ and $k \in \{0,1\}^{|\mathcal{R}|}$, where their elements are indexed by the index sets $\mathcal{S}$ and $\mathcal{R}$ respectively. The index sets satisfy $\...
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1answer
11 views

Poisson's Kinematical Equation in Index Notation

I'm trying to figure out the proper way to write Poisson's kinematical equation in index notation. The matrix form is $[\dot{C}] = -[\omega^\times][C]$ My first try is $\dot{C}_{jk}=-\epsilon_{...
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0answers
43 views

Reading off tensor index symmetries from a Young Tableau

I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form $[\mu][\nu]$ $[\rho][\sigma]$ I understand that if ...
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1answer
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bosonic interaction Heisenberg picture

I am trying to calculate the time evolution of the operator \begin{equation} h(k)=\sum_k b_k^{\dagger}b_k\, . \end{equation} Therefore, I go to the Heisenberg picture $$ h(k ,t) \equiv e^{\frac{i}{\...
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1answer
31 views

Mathematical equation/notation

How do I represent the following sum of products using summation notation? $$P = p_1 q_1 + p_2 (q_1+q_2) + p_3(q_1+q_2+q_3) + \dots $$ Here is my attempt: $P$ = $\sum_{i=1}^{n}{\{p_i\sum_{i=1}^{i}{...
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1answer
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Kronecker delta: in $3-$ dimension $\delta_{ii}=3$

My teacher said that in $3-$ dimension $\delta_{ii}=3$, but why? Kronecker delta's definition: $$\delta_{ij}=\begin{cases}0& \text{if}\; i\neq j \\ 1 & \text{if}\; i=j \end{cases}$$ ...
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1answer
32 views

Tensor Index Ordering

Whilst I agree that the order of tensor indices is important, $T_{ij} \ne T_{ji} $, I'm wondering if changing the order of the covariant and contravariant indices relative to eachother has any effect ...
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1answer
85 views

Proof of vector identity $(\nabla \times B)\times B = -\frac{1}{2}\nabla B^2 + (B \cdot \nabla)B$

I'm working through the book: Magnetohydrodynamics of the Earth's core (D. Gubbins, P.H. Roberts); J.A. Jacobs (Ed.), Geomagnetism, Vol. 2, Academic Press, London (1987) ... and I've come across the ...
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0answers
34 views

How does this sigma work?

May someone explains some first iterations of this sigma? https://i.stack.imgur.com/oh6bY.png Also, how did it convert the above expression to below expression? What it the meaning of I(x) and I(y)?
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0answers
61 views

Indices and powers of 2.

To find the value of $x^8$ given $x$, you need three arithmetic operations: $x^2=x\times x$, $x^4$=$x^2 \times x^2$ and $x^8=x^4\times x^4$. To find $x^{15}$, five operations will do: the first three ...
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1answer
28 views

Product of sums and loss of generality

I am struggling with seemingly simple algebraic manipulations with expressions containing finite sums. It is a physics based case but the interpretation of terms does not play any role now. Just note ...
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0answers
51 views

Writing out sum involving differentials and Levi-Cività tensor

I am currently working on an assignment involving the following expression: $$\frac16ε_{αβγδ}dx^βdx^γdx^δ=(dx^0dx^2dx^3,dx^0dx^1dx^3,dx^0dx^2dx^3,dx^1dx^2dx^3).$$ Now, I do not get how this equality ...
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170 views

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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0answers
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Specifying a range of indices belonging to a set

I have a set of different functions ordered according to their place in the set $\{1,2,\ldots,N\}$ and I want to specify a generic relation such that: $$\frac{\partial L_i(R_{i-j},\ldots,R_{i+k})}{\...
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0answers
17 views

Proving Contravariant Transformation with a Partial Derivative

I've been reading McConnell's Applications of Tensor Analysis, and in the first section on tensor analysis he gives the following problem. Show that $$\bar{a}^r = \frac{\partial \bar{x}^r}{\partial ...
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1answer
20 views

How to denote that two matrices have the same size, without using notation for the number of rows and columns?

How can I write that a matrix, e.g., $A_{p×q}$ has the same size as matrix $B_{m×n}$, i.e. $p=m$ and $q=n$, without using variables $p,q,m,n$, just the symbols for $A$ and $B$?
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Why in the product notation, if the top value is less than the bottom one, the product results to $1$? [duplicate]

As an example, for any one dimensional real valued function $f$, if $q<k$ then $\prod_{i=k}^q f(i)=1$. I thought it was $0$ and that was breaking my intuition about some formulas I'm working with. ...
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0answers
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Inverse of second rank rank tensor in index form

I have a second rank tensor (3*3 matrix) as follow: $$b_{{{\it ij}}}= \left( \lambda+\mu \right) k_{{i}}k_{{j}} $$ where i and j are indexes. The inverse for the above expression is presented as ...
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2answers
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What is the difference between $(u \cdot \nabla)v$ and $u\cdot(\nabla v)$ when written in einstein notation?

What is the difference between $(u \cdot \nabla)v$ and $u\cdot(\nabla v)$ when written in einstein notation? I understand that they are different, but I'm not quite sure how. I've proven $u \cdot (\...
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0answers
28 views

Correct notation for a matrix subset of an order-3 tensor?

Consider an order-3 tensor $\mathcal{T} \in \mathbb{R}^{n \times n\times n}.$ If I want to refer to a matrix "slice" of this tensor, what would be the conventional notation? For instance, in Matlab ...
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1answer
24 views

Auxiliary method to translate a (row,col) pair into a linear index

I don’t understand the meaning of the linind part of the code. Can someone explain it? ...
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2answers
32 views

Curl of $({\bf b}\cdot{\bf r}){\bf b}$?

I’m currently attempting to calculate $({\bf b}\cdot{\bf r}){\bf b}$. My attempt with index notation did not go far: $$\nabla\times({\bf b}\cdot{\bf r}){\bf b} = \epsilon_{ijk} \partial_j b_k b_l r_l....
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1answer
74 views

Index notation of double contraction with second order tensor derivative

I'm trying to wrap my head around an equation that involves the derivative of a second order tensor valued function of a second order tensor, then double-dot producted with another second order tensor:...
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1answer
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Index Notation Question

I wanted to know if someone could help me with this, please. This is my progress so far with the question: $N = 2\times 5^3\times x^4$ $N = 250x^4$ $N^2 = (250x^4)^2$ $N^2 = 62500x^8$ $5N^...
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1answer
18 views

Expressing the magnitude of a vector difference in indicial notation

I'm trying to express the following relation in indicial notation $$ |\vec{u} - \vec{v}_p| \, . $$ The only way I found out is replacing the difference above by $$ \vec{u} - \vec{v}_p = \vec{v}_r \, ,...
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0answers
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When to use subscript and when to use index?

I'm learning math and machine learning by myself, and I get confused when I see that some matrices are indexed by subscript whereas some are indexed by index. For example, the following Bellman ...
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0answers
61 views

Stokes' theorem in index notation

How do I derive the following expression for Stokes' theorem in index notation for tensors? $$\int_S\epsilon_{rst}\partial_tA_{jk\dots}dS=\oint_CA_{jk\dots}dx_r$$
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0answers
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Is there standard notation for a set of indexed families?

Let $\mathcal{I}$ be an index set, and $E$ another set. I have a set $A$ of tuples which I don't know how to express. Each contains both a subset of $\mathcal{I}$ ("domains") and a family of values ...
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1answer
32 views

Combination of an entry of a matrix in multiplicative form

Suppose $A$ is a $n\times n$ matrix with each entry $a_{ij}$. We all know the entry indexed by $(i,j)$ of $A^2$ would be $a_{ij}^{(2)}=\sum\limits_{k=1}^{n} a_{ik}a_{kj}$. My question is: Can we ...
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1answer
50 views

Abstract index notation vs Ricci Calculus

I have come accross some comparison between the abstract index notation and Ricci calculus as it pertains to contraction and what I find is: The former (abstract notation) indicates that a basis-...
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1answer
22 views

Diffusion on network

There is not any definition about $\delta_{ij}$ in my print for diffusion on network.What is $\delta_{ij}$? This is part of my print. $d\phi/dt = C(\sum_jA_{ij}\phi_j-\sum_jA_{ij}\phi_i)=C(\sum A_{ij}...
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0answers
48 views

Kronecker delta - substitution issues

I'm wondering if there are situations where index substitution using Kronecker deltas is not allowed? I'm currently fiddling with differentiation of the Softmax-function where I arrive at the ...
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2answers
33 views

Expression involving a power of 1.5 [closed]

How do you simplify u1.5 in root form? 0.5 is the root of u but im confused with 1.5 Please help Thankyou