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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.

508 questions
11
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1 answer

proving the determinant of a product of matrices is the product of their determinants in suffix / index notation

I'm trying to learn suffix notation (both to prove results in linear algebra and for application in vector calculus). As an exercise, I wanted to use it to prove that the determinant of a product of two matrices is equal to the product of their…
Rax Adaam
  • 1,265
8
votes
2 answers

Notation for direct sum of direct sums

Suppose you have a direct sum of two terms $A$ and $B$: $$ A \oplus B.$$ Now suppose that in fact $B = \bigoplus_i B_i$. Then I guess we could write the above sum as $$ A \oplus \bigoplus_i B_i.$$ However, this looks terrible -- much worse than, for…
rollover
  • 1,364
7
votes
2 answers

Structure coefficients of a coframe

From the textbook "Introduction to General Relativity, Black Holes & Cosmology" by Yvonne Choquet-Bruhat, p.10: In a moving coframe in a domain $U$, the differentials of 1-forms $\theta^i$ are given by: $$d\theta^i \equiv -\cfrac{1}{2} C^i_{jk}…
ElKorado
  • 183
6
votes
2 answers

Writing a double summation for distances between particles

I have a type of expression for a sum of distances between particles, which is as follows for four particles $$ d_{12} + d_{13} + d_{14} + d_{23}+d_{24} + d_{34}, $$ where $d_{12}$ is the distance between particles $1$ and $2$. I am trying to write…
Tom
  • 3,135
6
votes
1 answer

How to prove Leibniz rule for exterior derivative using abstract index notation

I want to prove Leibniz rule for exterior derivative of wedge product using abstract index notation: For $\omega\in \Omega^k(U),\eta\in\Omega^l(U)$, d$(\omega\wedge\eta)=\text{d}\omega\wedge\eta +(-1)^k\omega\wedge\text{d}\eta$. My proof is given…
Andrews
  • 4,293
6
votes
1 answer

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}{\hbar}Ht}\,\left( \sum_k b_k^{\dagger}b_k\right)\,…
mr. curious
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6
votes
2 answers

Einstein Notation

Evaluate $$\delta^{i}_{j}\delta^{j}_{i}$$when $1\leq i,j \leq n$ Simplify $$\delta^{a}_{b}g_{ca}g^{bd}\delta^{c}_{d}$$ when $a,b,c,d\in \{1,2,...,n\}$ in Einstein notation a matrix as a linear transformation is written as $$A=a^{i}_{j}$$ So…
gbox
  • 13,645
5
votes
3 answers

prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation

I'm having some trouble using index notation to prove the identity $$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$ The closest I can get is by expanding the first term on the RHS, which gives $$\mathbf{u…
user78655
  • 563
5
votes
2 answers

Abstract Index Notation Inconsistency (Technical - Answer in the Question)

I am reading this great work here and I am trying to make sense of a specific derivation around the middle of the page. In particular, it seems they are claiming that: $(\nabla_v w) (f) = (v^{\alpha} \nabla_{\alpha} w^{\beta})\nabla_{\beta}f$ where…
Pellenthor
  • 979
5
votes
2 answers

Prove that $\Gamma_{abc}=\frac{1}{2}\left(\partial_bg_{ac} + \partial_cg_{ab}-\partial_ag_{bc}\right)$

I am tasked with the following problem Use the equation $$\nabla_ag_{bc}=\partial_ag_{bc}-\Gamma_{cba}-\Gamma_{bca}=0\tag{1}$$ where $$\Gamma_{abc}=g_{ad}\Gamma^d_{bc}\tag{A}$$ and the (no torsion) condition $$\Gamma_{abc}=\Gamma_{acb}\tag{i}$$ to…
user1017949
5
votes
1 answer

Confusion in Notation for Array and Dimension

I have a vector $\mathbf{x} = (x_1,x_2)$, how do I represent the array of such vector? Is it $\mathbf{x} \in \mathbf{X}$ where $\mathbf{X}$ = $\bigcup _{{1}}^{k}\mathbf{x}^s$ for $s \in (1,..k)$ length of $k$ vectors? Something seems off here! Also…
user0193
  • 313
5
votes
1 answer

Sigma index (Induction)

I was wondering how the index that's left is measured. When you prove this via induction, $$\sum_{k=1}^{2^n} \frac{1}{k} \geq \frac{n}{2} $$ $$\sum_{k=1}^{2^{n+1}} \frac{1}{k}= \sum_{k=1}^{2^{n}} \frac{1}{k} + \sum_{k=2^n+1}^{2^{n+1}} \frac{1}{k}…
Seenes
  • 87
5
votes
1 answer

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 of this post. The equation is called the "Bianchi…
gurfle
  • 103
5
votes
1 answer

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 on the tensors, in other words; are the following…
user2662833
  • 2,046
4
votes
1 answer

Is the index notation $1 \leq i \neq j \leq k$ unambiguous?

From the context where I found the statement $$1 \leq i \neq j \leq k,$$ the author seems to mean that $1 \leq i \leq k$ and $1 \leq j \leq k$ and $i \neq j.$ However, strictly speaking, the given statement doesn't imply that $i \leq k.$ For…
Kuku
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