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Mathematics Stack Exchange

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.

For questions about index notations, including 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.

488 questions
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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 if all the components shared the same value?…
The Shepard
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How to convert an einsum expression to an equation?

I'm using Python's einsum for an operation over two arrays, and I was wondering how to correctly write this operation out in a paper. Let's say I have two '3d' arrays $A_{i,j,k}$ and $B_{i,k,l}$, and I want a sum-product over index $k$, an outer…
thepainofstats
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Einstein Notation Of An Inverse Matrix

When we look at a matrix $A$ as a linear map, we write the element of the matrix as $a^{i}_{j}$ so the inverse matrix will be? In the case of bilinear form for $a_{ij}$ the inverse is $a^{ij}$ or $a^{ji}$?
gbox
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Question on Einstein Summation convention evaluating certain expression

Consider the following expression: $$X^k\frac{\partial g_{ij}}{\partial x^k} + g_{jk}\frac{\partial X^k}{\partial x^i} + g_{ik}\frac{\partial X^k}{\partial x^j} = 0$$ Am I correct, that this is the same as $$\sum_k \left(X^k\frac{\partial…
TheGeekGreek
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Showing that a function is a Fourier Multiplier (Umarov)

Hello. In example 1.8.2, it is shown that $|t^mp^{m}(t)|\leq C$ for all $t>0$ and finally, the function $m(\xi)$ satisfies the condition 1.68. In my case, I am playing with the function $\rho(t)=t/(1+t)$ ($\gamma=1$) along with $n=2$ (i.e.…
eraldcoil
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How to treat partial derivatives to prove this identity using index notation?

I'm trying to verify this identity using index notation: $$ \vec{V} \times \left(\nabla \times \vec{V}\right) = \frac{1}{2} \nabla\left(\vec{V}\cdot \vec{V}\right)-\left(\nabla\cdot\vec{V}\right)\vec{V}$$ Solving the LHS: $$v_i \hat{e_i} \times…
J. J. Gonzalez
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Disambiguation of numerical base and index variable in mathematics

Is there a proper way to distinguish base from index variable? For this instance, let's say we have a paper that deals with cryptographic keys both in different bases and with different indices. For different bases we might say that our binary…
mcp
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Levi-Civita symbol with matrix

What is the matrix form of $\epsilon_{ijk} A_{jk}$?
user157226
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Is it possible to convert output of an equation in an unordered set?

I am total novice in algebra, so I need help regarding what I try to do... I have built the following equation: and these three statements: Here is what is the process I tried to describe: B is a family set of length 21, containing rounded…
RPO
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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{u} \vec{v}) = (\nabla \cdot \vec{u})\vec{v}…
Stone Preston
  • 139
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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: $P$ = $\sum_{i=1}^{n}{\{p_i\sum_{i=1}^{i}{q_i}\}}$ where $i = 1,2,...n$ Thanks Note: what I am…
hkf
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How to decide whether an equation in index notation is valid.

I am given the following equation in index notation: $k_{ijkl} = a_{i}b_{kl}c_{njm}d_{mn} + e_{ik}e_{jn}f_{n}$. I am told that this is a valid equation, but can anyone explain why? It doesn't violate the summation convention, and there's no obvious…
wrb98
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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-operator, $\boldsymbol{\nabla}$. I know that in three…
Rasmus Andersen
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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…
quantumNeko
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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: $\nabla(A \cdot B)=A \times (\nabla \times B)+(A \cdot \nabla)B+B \times (\nabla \times A)+(B \cdot \nabla)A$ $\partial_i(a_j b_j)$…
physicsrocks
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