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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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Conversion of mixed tensors into mixed tensors and into covariant (or contravariant) ones

I am an undergraduate student of Physics, currently taking a course on Special Relativity, but I am getting too confused with tensors and their indices. My question is: How to convert mixed tensors to ...
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1answer
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Trouble with changing indexing in summation

I have trouble with the following: $$P(t) = \sum_{i=0}^{n-r}c_i^r(tB^{n-r-1}_{i-1}(t)+(1-t)B_i^{n-r-1}(t))$$ $$= \sum_{i=0}^{n-r-1}(tc^r_{i+1}+(1-t)c_i^r)B_i^{n-r-1}(t),$$ where $c_i$ are constants, ...
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Calculate the characters from a combination

Given a table with position - 2 character combination pairs like this: 1. aa 2. ab 3. ac 4. ad ... 27. ba 28. bb 29. bc 30. bd Assuming there are unique ...
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How to derive the non-torsion-free Bianchi identity by building a canonical torsion-free alternate derivative from the original covariant derivative

I know the Bianchi identity can be derived much more directly and simply (as is apparent in this post). The point here is to follow this alternate path to it as proposed in the paragraph after ...
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Order of indices using index notation

I'm trying to solve a problem in Lagrangian mechanics involving index notation. I'm wondering if the expression $$ A_{ij}\dot{q^i}q^j = A_{ji}\dot{q^j}q^i $$ is true. Our professor skipped over this ...
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1answer
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Using diagonality in Einstein notation

Given a diagonal matrix $D$, with diagonal elements given by vector $\mathbf{d}$. Representing this in Einstein notation gives $$ D_{ij} = \delta_{ijk} d_k $$ where $$ \delta_{ijk} = \begin{cases}...
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How to prove Penrose “Bianchi symmetry” with non-zero torsion tensor using abstract indexing?

I want to prove $R_{[αβγ]}^{\ \ \ \ \ \ \ δ} + ∇_{[α}T_{βγ]}^{\ \ \ \ δ} + T_{[αβ}^{\ \ \ \ ρ}\ T_{γ]ρ}^{\ \ \ \ δ} = 0$ EDIT: A brief discussion of the solution found by Matt is at the bottom ...
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Changing index of summation

How do you go from $\sum_{n=0}^{\infty} z^{-n-1}$ to $\sum_{n=-\infty}^{-1} z^{n}$ ?? It's really confusing.
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1answer
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Expressing the coordinate dependent and indepent forms of the $(0,1)$ tensor in different coordinate systems

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
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Transformation rule in differential geometry

I am reading Walds General Relativity and am looking at Question 8, Chapter 2. In the solutions to this question it states that the metric is determined by the transformation rule $$g_{\alpha\beta}^{'...
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Show that $\nabla^{2}\left ( x_{i}x_{j}r^{n} \right ) = 2\delta _{ij}r^{n}+n\left ( n+5 \right )x_{i}x_{j}r^{n-2}$

No matter what I can't seem to arrive at this answer. I've tried $\partial_{i}\left (\partial_{i}\left ( x_{i}x_{j}r^{n} \right ) \right )$ and $\delta _{ij}\partial_{i}\left (\partial_{j}\left ( ...
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1answer
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Ordinary Least Squared derivation using Index Notation

The following function in $\mathbf{x}$ is given: $$f(\mathbf{x}) = \frac{1}{2}\left\lVert \mathbf{Ax - b} \right\rVert_2^2$$ I want to calculate $\frac{\mathrm{d}f(\mathbf{x})}{\mathrm{d}\mathbf{x}}$...
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Prove equality with cross products using Einstein notation

Notice that throughout this question I will be using Einstein's summation notation. Let $v(x):\mathbb{R}^3 \to \mathbb{R}^3$ be a vector field and define $$2\mathbb{W}(v) = \nabla v - \nabla v^T$$ ...
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Validity of Index Notation

I have the following questions on my biomechanics homework (we're finally getting to multidimensional spaces). However, we haven't gone over it yet and won't have class again until after it's due. I'...
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divergence of dyadic product using index notation

I am trying to prove the divergence of a dyadic product using index notation but I am not sure how to apply the product rule when it comes to the dot product. I would like to show: $\nabla\cdot (\vec{...
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Show that the laplacian of the curl of A equals the curl of the laplacian of A. $\nabla^2(\nabla\times A) = \nabla \times(\nabla^2A)$

I might have worded the question incorrectly in the title, but I am trying to show that $\nabla^2(\nabla\times \vec A) = \nabla \times(\nabla^2 \vec A)$. I am not sure if there is any physical ...
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Tensor and Vector Summation Convention using Index Notation

A second-rank tensor A and a vector v are given by: A= $\begin{bmatrix}1&0&1\\0&2&0\\2&0&3\end{bmatrix}$ v= $\begin{bmatrix}1\\0\\1\end{bmatrix}$ Taking into account the ...
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1answer
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Need help understading prime notation for vector/time series

I have vector/time series with ' symbol at the end: X=(x1,x2,...,xn)' What does it mean? It is derivative of vector/time series?
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3answers
51 views

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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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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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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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
34 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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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
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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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38 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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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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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
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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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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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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
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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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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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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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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
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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
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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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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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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
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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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256 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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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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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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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 $B$?