# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $j=1,\dots,J$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### Usage of dummy indices in a sum

A current textbook I'm reading has the following example of how to take derivatives of summations $$\frac{d}{dt}\sqrt{\sum_jA_jB_j} = \frac{\sum_k\dot{A}_kB_k + A_k\dot{B}_k}{2\sqrt{\sum_jA_jB_j}}$$ ...
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### Matrix manipulations with Levi-Civita symbol

My question relates to this reply on math.stackexchange. More precisely, I am wondering about the following sequence of expressions involving elements of an invertible square matrix $M$ and a pair of ...
1 vote
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For a contravariant basis vector in ${\Bbb R}^n$ that is defined using a covariant basis vector ${\bf Z}_j$ in ${\Bbb R}^n$ in terms of the contravarient metric tensor as: $${\bf Z}^i = Z^{ij}{\bf Z}... 0 votes 1 answer 41 views ### How to prove this equality using Einstein summation (index notation) I have to prove this equality using index notation (Einstein summation), but I don't know how to proceed from here: \nabla(A \cdot B)=A \times (\nabla \times B)+(A \cdot \nabla)B+B \times (\nabla \... -1 votes 1 answer 57 views ### Could you help me with this problem on natural numbers? Pick two natural numbers q,n \in \mathbb{N} such that 1<q<n and for 0 \leq j\leq q-1 define a(j) = \lfloor \frac{n-j}{q}\rfloor. The claim is that the elements of the set \{rq+j \mid 0\... 2 votes 1 answer 64 views ### Is \int_{-\infty}^\infty \sum_{g(x)=a}\frac{f(a)}{|g’(x)|} da=\left(\frac1{|g’(a_1)|}+\dots+\frac1{|g’(a_j)|}\right)\int_{-\infty}^\infty f(a)da? In A strange integral: \int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi. @robjohn posts that:$$\int_{—\infty}^\infty f(g(x))dx=\int_{-\infty}^\infty \sum_{g(x)=a}\frac{f(a)...
In this video on Tensor calculus(timestamped), the professor explains how the matrix entry is written in a tensor-like notation: $$\text{(i,j)}^{th}\text{ entry of matrix A}=A_{ij}=A_j^i$$ Later in ...