Questions tagged [index-notation]

Question 1

I came across the following identity in 2D (and I believe in 2D only): $$\varepsilon_{kj}\delta_{il}-\delta_{ij}\varepsilon_{kl}=-\varepsilon_{jl}\delta_{ik},$$ with $\varepsilon$ the Levi-Civita ...

Question 2

I wish to establish a consistent notation for my work, which involves combinations of different attributes of a variable $\boldsymbol{x}$ given in the following, including the notation most common in ...

Question 3

The mixed partial derivative $\frac{\partial^{2} f}{\partial x \, \partial y} := \frac{\partial}{\partial x}\!\left( \frac{\partial f}{\partial y} \right)$ is obtained by first differentiating with ...

Question 4

I have the following problem. I have a set of orders $i\in I=\{1,2,...,|I|\}$. Each order has three properties $A_i,B_i,C_i$. Now I want to aggregate those orders with the same properties into an ...

Question 5

Following the paper " WAVEPACKETS IN INHOMOGENEOUS PERIODIC MEDIA: EFFECTIVE PARTICLE-FIELD DYNAMICS AND BERRY CURVATURE" by Watson, i want to convince myself that the Ansatz \begin{equation}...

Question 6

For $n \in \mathbb{Z}_+$, $\alpha \in \{0, 1\}$, $\langle x_q \rangle_{q=1}^n \in \mathbb{R}^n$ and $f : \mathbb{R} \rightarrow \mathbb{R} $, let us consider the following possible expressions for a ...

Question 7

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 ...

Question 8

In index notation, the determinant of a 3 by 3 matrix $A$ is $\det(A)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}$, and the product of matrices $A$ and $B$ is $(AB)_{ij}=A_{ik}B_{kj}$. Thus, the determinant of ...

Question 9

In almost all places I look, the generalized Kronecker-delta symbol is defined as: $$\varepsilon_{i_1...i_p}\varepsilon^{j_1...j_p}=\delta _{j_1...j_p}^{i_1...i_p}$$ Which is the way I know it to be ...

Question 10

Is it possible to create a rank 3 tensor that is skew-symmetric in one pair of indices but symmetric in another pair? That is, a tensor whose components satisfy $$S_{ijk}=-S_{jik}=S_{kji}$$ How many ...

Question 11

I have this equation, $$\frac{\partial\phi_i}{\partial \mu_j}\mu_j = T_{i,j}\frac{\partial\phi_j}{\partial \mu_k}\mu_k + g_i$$ Could I use identity matrix to change the indices so that I can rewrite ...

Question 12

In this proof (source: https://www.math.toronto.edu/aretakis/General%20Relativity-Aretakis.pdf, pag 123) Substituing $ g^{-1} (\nabla^\mu d\psi, d\psi) \cdot g_{\mu\nu} = (\nabla^b d\psi)_{\nu} \...

Question 13

I am a beginner with index-notation. I'm having trouble with specific examples of finding the gradient of a vector. Assume $\partial_i = \frac{\partial}{\partial x_i}$. Generally, the following basic ...

Question 14

I was doing problem 2.11 from the book Continuum Mechanics for Engineers (4th Ed.) and I just can't figure out how to solve this problem without filling out two whole whiteboards. I know there's got ...

Question 15

When I have a cross product of two vectors a and x and I want to square it, I write (in index notation): $$ (\textbf a\times\textbf x)=\varepsilon_{ijk} a_jx_k\varepsilon_{ilm}a_lx_m $$ $$ =(\delta_{...

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Questions tagged [index-notation]

How to derive Levi-civita identity: $\varepsilon_{kj}\delta_{il}-\delta_{ij}\varepsilon_{kl}=-\varepsilon_{jl}\delta_{ik}$ in 2D

Notation for combination of indexing, iterating, and labeling of variables

Index notation for mixed partial derivatives – $f_{xy}=(f_x)_y$ or $f_{xy}=(f_y)_x$?

How to formally define aggregated index sets based on shared attributes?

Contraction/swapping of indices

Question on Einstein Notation for Distinct Expressions

Writing a double summation for distances between particles

How to prove $\det(AB)=\det(A)\det(B)$ in index notation? [duplicate]

Generalized Kronecker delta and metric sign

Rank-3 tensor that's skew and symmetric on different pairs of indices

Using Identity Matrix to Change Indices in Einstein Notation

Why is $(\nabla^a d\psi)_{\mu} \cdot (d\psi)_{\nu} = (d\psi)_{\mu} \cdot (\nabla^a d\psi)_{\nu} = (\nabla^b d\psi)_{\nu} \cdot (d\psi)_b$?

Failing with dummy index notation when computing $\nabla_u \exp (\overrightarrow u)$

How do I prove this concisely with indicial notation? [closed]

Indicial notation of square of cross product

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