Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
34 views

Is this a valid notation for a directional derivative?

In Siegel (2005): Fields I found the following strange notation on page 170 $$ \frac{\partial f(\phi_i)}{\partial \phi_j} = \lim_{\epsilon\to0}\frac{f(\phi_i + \epsilon \delta_{ij})-f(\phi_i)}{\...
asmaier's user avatar
  • 2,674
0 votes
0 answers
16 views

Incrementing Indices of Variables through some kind of Operator

I have some function which contains variables $B_i$ and want a way to transform this function into the same operations of the variable but just with the indices shifted: $B_i \rightarrow B_{i+1}$ For ...
Kiwi's user avatar
  • 1
2 votes
2 answers
73 views

Notation Dilemma in Formulas

Considering: 1. $$ w^{(k+1)} = w^{(k)} - \eta g_{w^{(k)}}$$ Given this, do I need to define $g_{w^{(k)}}$ with the index?: 2. $$ g_{w^{(k)}} = \frac{1}{m} \sum_{i=1}^m \nabla_{w^{(k)}} L(h(x_i, w^{(k)}...
slaky's user avatar
  • 21
0 votes
1 answer
94 views

Abstract index notation - can't understand identity $(dx^{\mu})_aT^b\partial_bv^a=T^b\partial_b[(dx^{\mu})_av^a]$

I'm trying to get familiar with abstract index notation. Came across this identity: $$(dx^{\mu})_aT^b\partial_bv^a=T^b\partial_b[(dx^{\mu})_av^a]=T^b\partial_bv^{\mu}$$ where $T$ and $v$ are vector ...
Shirish's user avatar
  • 2,569
0 votes
0 answers
35 views

Sum over an index that needs to fulfill a condition

Introduction I have a matrix $V=(v_{ij})_{1\leq i \leq m, 1\leq j \leq n}$ containing integer numbers, representing the number of vehicles. Each $i-$row represents the place of origin of those ...
Ommo's user avatar
  • 349
0 votes
1 answer
46 views

mathematical notation of a product of n-uplet elements

I have the following $n$ sets: $A_{1}=\{A_{1}^{1}, A_{1}^{2}, ... , A_{1}^{j_1}\}$ ; $A_{2}=\{A_{2}^{1}, A_{2}^{2}, ... , A_{2}^{j_2}\}$ ; ... $A_{n}=\{A_{n}^1, A_{n}^{2}, ... , A_{n}^{j_n}\}$. Let $m=...
Ft insat's user avatar
0 votes
1 answer
47 views

Expand product of sums in sympy by introducing new dummy indices

I want to use sympy to simplify some expressions which contain products of sums, this will require expanding out the products and cancelling equal terms. ...
cyfirx's user avatar
  • 35
1 vote
0 answers
22 views

Simplifying After Using the Bloch Ansatz in the Stationary Schrödinger Equation

I am working on deriving the intensity equations for the dynamical diffraction of neutrons following along with a paper by Hartmut Lemmel (Hartmut Lemmel. Dynamical diffraction of neutrons and ...
pyguy83's user avatar
  • 11
0 votes
0 answers
10 views

Multi-sigma functions with unequal indices

I have a faint memory of having read about general considerations regarding the evaluation of sums of the form: $$\sum_{i_1=1}^{U_1}\sum_{i_2=1 \\ i_2 \ne i_1}^{U_2} \cdots\sum_{i_k =1 \\ i_k \ne i_{j&...
user110391's user avatar
  • 1,129
0 votes
0 answers
11 views

Terminology/notation for pulling back multi-sigma functions

I deal with a lot of multi-sigma functions, which I assign compact formulas through a step-wise "pull-back". It is best explained through a well-known example: $$\begin{align} \sum_{i_1=1}^n ...
user110391's user avatar
  • 1,129
0 votes
0 answers
53 views

Inverse tensor notation

I have been learning how to transform from one basis to another. I don't have any issues when both basis are orthogonal because I use a different formula then the one below. An issue I am having is ...
Alexander Savadelis's user avatar
1 vote
2 answers
200 views

Inversion of a matrix equation

Is there a general way to invert (solve for $u$) this? $$\sum_{ij}R_{ijk}a_iu_j = -x_k$$ With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices. So really I'm trying to invert ...
Gennaro Marco Devincenzis's user avatar
0 votes
1 answer
36 views

Geometric series with indexed inequality

I wanted to complete the following sum: $$\sum_{0\leq i <j<k} a_ib_jc_k$$ Where $a_ib_jc_k$ are all different geometric sequences with $|r|<1$. My attempt was to break up the sum into what ...
beigespectacles's user avatar
3 votes
1 answer
48 views

Commutator formula between Hessian and Laplacian of a scalar function

I am looking to derive an identity I found which commutes the Hessian and the Laplacian of a scalar function $f \in C^4(M)$ on a Riemannian manifold $(M,g).$ $$\Delta \nabla_i \nabla_j f = \nabla_i \...
DrHAL's user avatar
  • 865
0 votes
0 answers
39 views

Changing indexing

This is probably rather trivial, but anything that's combinatorics related messes me up. Just to check: are the following summations identical? $$\sum_{k_1+k_2+k_3 = n} a^{k_1} b^{k_2} c^{k_3} \quad \...
OtherQuestions's user avatar
3 votes
1 answer
106 views

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 ...
digital's user avatar
  • 185
3 votes
1 answer
220 views

Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$

The following is loosely related to this question: [...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is $$\mathcal{L}=-\...
Sirius Black's user avatar
2 votes
1 answer
136 views

Notation for the d'Alembert operator

From the Wikipedia page on the d'Alembert operator it is stated that equivalent ways of writing the d'Alembert operator are as follows, $$\begin{align}\Box =\eta^{\mu\nu} \partial_\nu\partial_\mu = \...
Sirius Black's user avatar
1 vote
1 answer
182 views

Using index notation to prove vector identities

I am asked to use index notation to prove that $\nabla\cdot(\overrightarrow{u}\times\overrightarrow{v})=(\nabla\times\overrightarrow{u})\cdot\overrightarrow{v}-\overrightarrow{u}\cdot(\nabla\times\...
End points's user avatar
4 votes
1 answer
97 views

Hermitian adjoint for 3-vectors; should the energy-momentum 4-vector be written as $P^\mu=\left(p^0,\,p^i\right)$ or $\left(p^0,\,\vec {p}\right)$?

Consider the real Klein-Gordon scalar; Taking the Hermitian Hamiltonian, and (spatial) momentum operators as $$\hat H = \int \frac{d^3 \vec p}{\left(2 \pi\right)^32E(\vec p)}E(\vec p)\hat a^\dagger(\...
FutureCop's user avatar
  • 237
2 votes
0 answers
27 views

Evaluate number of terms in a double summation

I am studying tensors and I have an exercise in finding the number of independent values of a symmetric (2,0)-tensor $B^{IJ}$, where $I, J \in$ {1,2,3, ..., d}. Since the entry of $B^{IJ}$ is the same ...
daprowe's user avatar
  • 21
1 vote
1 answer
122 views

Einstein notation and differential operators.

So I am dealing with the differential operator $\mathbf{D} = \mathbf{r} \times \boldsymbol{\nabla}$ where $r = x_i \mathbf{e}_i$. We then introduce $(\mathbf{D}f)_i$, which can be expressed as $\...
user1250010's user avatar
0 votes
0 answers
37 views

Indexing finite subsets

Let $X\triangleq\{x_1,\dots,x_n\}$ be a set of $n$ vectors $x_1, \dots, x_n \in\mathbb{R}^d$. I'm trying to formalize how to index all possible subsets $Y\subseteq X$ because I have to compute a ...
matteogost's user avatar
0 votes
0 answers
31 views

Index notation for Gateaux derivative of matrix function confined to subspace of traceless, symmetric matrices

Suppose I have a function $\Lambda: S^\text{tr} \to S^\text{tr}$ where $S^\text{tr} = \{Q \in \mathbb{R}^{n \times n} \: |\: Q = Q^T \: \text{and} \: \text{tr}(Q) = 0\}$ is the space of traceless, ...
Lucas Myers's user avatar
1 vote
0 answers
58 views

Notation about the size of a subset

Introduction. I have three indices, $i$, $j$ and $l$ indicating, respectively, the $i^{th}-$element, the $j^{th}-$element and the the $l^{th}-$element of a set of 2-dimensional points $P=\{ p_1, p_2, ...
Ommo's user avatar
  • 349
2 votes
0 answers
86 views

Mathematical definitions of in-degree and out-degree in a graph

Introduction. I have a kNN graph, which is a directed graph where each node $i$ has out-degree $k^{out}_i$. The out-degree $k^{out}_i$ is equal for every node $i$, and it is equal to the number of $k-...
Ommo's user avatar
  • 349
0 votes
0 answers
105 views

Is the inverse of the metric tensor also a tensor?

In my textbook the inverse of the metric tensor ($g^{ij}$) is defined the following way: $$g^{ij}g_{jk}=\delta^i_k$$ Then if we have the matrices forward and backward transforamtion $A$ and $B$, we ...
Krum Kutsarov's user avatar
0 votes
0 answers
17 views

join conditions for subsequences

Introduction. We consider a sequence $a_{1},a_{2},\cdots,a_{n}$ and two subsequences. One subsequence is $a_{i,m}$, of length $m$ and starting at $i$, while the second subsequence is $a_{j,m}$, of ...
Ommo's user avatar
  • 349
2 votes
0 answers
52 views

What is the proper way to typeset a vector that has a uniform value?

I have an first-rank tensor for acceleration in four dimensions. My first instinct would be to write that out as $A^\alpha$ to show there are four components to the value, but how would you write this ...
The Shepard's user avatar
0 votes
0 answers
59 views

Summation with inner products: properties and rearrangement

OPTION 1. I have this expression, $$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$ and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
CafféSospeso's user avatar
0 votes
0 answers
49 views

Tensor Index Manipulation

I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that $\partial_{\mu} g^...
 Paranoid's user avatar
0 votes
1 answer
23 views

Evaluation of an indexed sum. [closed]

I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds. $\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$ This is very simple I know, but ...
Newbie1000's user avatar
1 vote
1 answer
54 views

What formula can I use to translate $m$ indices into $n$ indices?

NOTE: The tensors are considered row-major. The sequence of elements is preserved. Let $A$ be a $m$-dimensional tensor of size $m_1 \times m_2 \times m_3 \cdots$ and $B$ be a $n$-dimensional tensor of ...
user366312's user avatar
  • 1,671
0 votes
1 answer
97 views

Derivative of block matrix using einstein notation

Let $Y = [A \quad XB \quad C]$, where $A,B,C,X$ are all matrices with appropriate size. What is the derivative of $Y$ w.r.t. $X$? The part that confuses me is that $\frac{\partial A}{\partial X}$ ...
nku's user avatar
  • 1
0 votes
1 answer
21 views

Indexing and $∪_{p∈P}C_p = (∪_{i∈I} A_i)\times (∪_{i∈I} B_i)$

The following is a problem question in How to Prove It (4.1.14b): For each $(i, j) ∈ I × I$ let $C(i,,j) = A_i × B_j$, and let $P = I × I$. Prove that $∪_{p∈P}C_p = (∪_{i∈I} A_i)\times (∪_{i∈I} B_i)$ ...
airwick's user avatar
  • 63
0 votes
0 answers
91 views

Confusion about Levi-Civita determinant identity

From the wiki page on the Levi-Civita symbol, they write the following equation $$ \sum_{i_1,i_2,...}\epsilon_{i_1...i_n}a_{i_1j_1}...a_{i_n j_n}=\det(\mathbf{A})\epsilon_{j_1...j_n}, $$ which I ...
Forum's user avatar
  • 15
0 votes
1 answer
316 views

What do the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ mean?

I am reading David Tong's notes on vector calculus (which are amazing), and the symbols $\delta_{ij}$ and $\epsilon_{ijk}$ keep coming up in the notes. I do not know what these actually mean. I think ...
wlancer's user avatar
  • 13
0 votes
1 answer
70 views

Question regarding the index notation: Interpretation of $\partial_n r_k$

I have recently been studying and using the index notation in physics, but I have a specific question, which is not very clear to me. Say we have the radial vector $\textbf{r}$ and the usual Del-...
Rasmus Andersen's user avatar
1 vote
2 answers
147 views

Proof of this identity without using index notation

I want to see the following identity that is used in the proof of the first version of Bernoulli's theorem without using index notation $(\mathbf{u} \cdot \mathbf{\nabla})\mathbf{u} = (\nabla \times \...
Franco Medina's user avatar
1 vote
0 answers
63 views

Notation ${}_{\mathcal B}[h]_{\mathcal A}$ or $[h]_{\mathcal A}^{\mathcal B}$ for the matrix of a linear map $h$

I've seen a few times the notation ${}_{\mathcal B}[h]_{\mathcal A}$ for the matrix of a linear map $h\colon E\to F$ with respect to a base $\mathcal A$ of $E$ and a base $\mathcal B$ of $F$. I ...
Alexey's user avatar
  • 2,210
3 votes
1 answer
139 views

How to prove $\det{AB}=\det{A}\det{B}$ with Leibniz formula in terms of Levi-Civita symbol and Einstein summation notation? [closed]

Prove that $\det{AB}=\det{A}\det{B}$ with Leibniz formula in terms of Levi-Civita symbol and Einstein summation notation Here is a similar question asked 6 years ago. The OP answered in the question ...
Ryoko Asakura's user avatar
0 votes
1 answer
112 views

Compact notation for multiple sums

Consider a multiple series of the form: $$ \sum_{k_1=0}^\infty\cdots\sum_{k_n=0}^\infty f(k_1,\dots,k_n). $$ I have a more complicated multiple series where the indicies are $k_{jn}$ with $j=1,\dots,J$...
Aaron Hendrickson's user avatar
0 votes
1 answer
72 views

Possibility (generality) of symmetrization wrt certain indices

Consider the following contraction between two vector fields $$ A_{k,i}B_{k,j} $$ Summation over $k$ is implied. I want to decompose this into parts that are symmetric/antisymmetric w.r.t the ...
Bjaam's user avatar
  • 77
0 votes
1 answer
265 views

Deriving the coordinate formula for the Gauss curvature under conformal map using the moving frames method

As an exercise, I wanted to derive this formula for the Gaussian curvature (with n=2) under a conformal map: $$\tilde{K}=e^{-2\rho}K-e^{-2\rho}\Delta\rho$$ with the method of moving frames. Given two ...
ebenezer's user avatar
  • 121
1 vote
0 answers
97 views

Equivalent formulas for Laplace-Beltrami operator

I am trying to derive the following equivalence: $$ g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j}\frac{\partial f}{\partial x^k}+ \frac{1}{2}g^{jk}g^{il}\...
ebenezer's user avatar
  • 121
3 votes
2 answers
182 views

Index Notation - An Inconsistency with a deeper meaning?

Background Suppose we have a linear transformation $T : V \to W$ where $V$ and $W$ are finite dimensional vector spaces with bases $(e_i)$ and $(f_i)$ respectively. Using index notation, it is ...
Charles Hudgins's user avatar
0 votes
1 answer
174 views

How to write expressions in tensor notation?

I’m a self-taught math student and i want to learn how to work and calculate with tensors. I’m pretty familiar with the theory behind them, but something about the the different notations just does ...
Simón Flavio Ibañez's user avatar
0 votes
1 answer
40 views

Index Notation Simplification Help

Given a real matrix $F$, I'm trying to simplify the expression $$ C_{ijkl} = \frac{\partial (F_{ip}F_{jp})}{\partial F_{km}} F_{lm}$$ One reference I saw online says the right-hand side simplifies to: ...
asyndeton256's user avatar
0 votes
0 answers
35 views

Why Does the Trace of the following Double Derivative Expression Vanish?

I need some help completing the last part of this problem. Problem: For the position vector $r$ such that $r = x^2 + y^2 + z^2 = \sum_{i=1}x_i^2$, show that the trace of the following expression $$\...
quantumNeko's user avatar
1 vote
1 answer
77 views

Identity tensor outer product representation

There is an equation (from Chapman-Enskog theory) that uses index notation: $$\sum_i w_ie_{i\alpha}e_{i\beta}e_{i\gamma}e_{i\mu}= c_s^4(\delta_{\alpha\beta}\delta_{\gamma\mu}+ \delta_{\alpha\gamma}\...
Chunheng Zhao's user avatar

1
2 3 4 5
10