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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.

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28 views

Is this matrix notation correct?

I have a bunch of matrices that I denote as follows: $$\mathbf{A_j}=(a_{c,k}^j)\in[0,1]^{d\times q}$$ The reason I don't use a tensor of order $3$ is because $d$ will vary for each $j$, which is why ...
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1answer
24 views

d’Alembert Lagrangian for second rank tensors

Consider the Lorentz scalar Lagrange density $$\mathcal{L}=\eta^{\mu\nu}\partial_\mu T^{\alpha\beta}\partial_\nu T_{\alpha\beta}$$ for a second rank tensor whose contravariant and covariant ...
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1answer
16 views

Index notation problem in getting $(\Lambda^{-1})^{\mu}_{\nu}=g^{\mu\alpha}g_{\nu\beta}\Lambda^{\beta}_{\alpha}$

Let consider: $g_{\mu\nu}=diag(1,-1,-1,-1)$ and $\Lambda$ so that $\Lambda^{T}g$$\Lambda$=g I want to prove that $g=g\Lambda\Lambda \Rightarrow \Lambda^{-1}=g^{-1}g\Lambda$ The proof is silly for ...
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1answer
57 views

Differentiation of tensor product

I have a tensor equation $$\frac{\partial A_{ij}}{\partial B_{kl}}=\frac{\partial A_{ij}}{\partial C_{pq}}\frac{\partial C_{pq}}{\partial B_{kl}} $$ $C_{pq}$ can be written as $C_{pq}=B_{pq}+aB_{mm}\...
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0answers
31 views

When do derivatives cancel inside integrals when working with tensors?

While doing a problem recently I realised I'm not clear about when derivatives inside integrals will cancel when working with tensors. For example, I have come across integrals such as: $\int \...
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2answers
73 views

divergence of gradient of scalar function in tensor form

I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function: $$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
4
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1answer
91 views

Use abstract index notation to prove Leibniz rule for exterior derivative

I want to use abstract index notation to prove Leibniz rule for exterior derivative of wedge product: For $\omega\in \Omega^k(U),\eta\in\Omega^l(U)$, d$(\omega\wedge\eta)=\text{d}\omega\wedge\eta +(...
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0answers
13 views

Specific, Index-Notation Formula of the determinant of 2-dimensional matrices

I was practicing some old tensor calculus exams and had to answer this question (paraphrased, it has multiple subquestions): If $\eta_{ij}$ is the component of the metric $H$ on a 2D Riemannian ...
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0answers
18 views

Multivariate Gaussian distribution notation

Is the following correct? The prior distribution is a multivariate Gaussian distribution defined as: \begin{equation} \boldsymbol {\mathrm {\theta}}_i \sim \mathcal{MVN}( \boldsymbol {\mathrm {\mu}},...
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1answer
67 views

Prove derivative of contravariant tensor of rank 1 is a mixed tensor of rank 2

$A^\alpha$ is a given contravariant vector (when $\alpha\in {0,1,2,3}$ a $4$-vector in Minkowski space) I need to show that the derivative $\frac{\partial A^\alpha}{\partial x^\beta}$ is a mixed ...
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1answer
70 views

How do multi-indices work, step by step?

Never used multi-indexed summations in my life, neither has anyone else I know. https://en.wikipedia.org/wiki/Multinomial_theorem does not define an upper index for the multi-indexed sum, which ...
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1answer
23 views

Are “lower bound” and “upper bound” unambigous? What if one's in the negative axis?

Are "lower bound" and "upper bound" unambigous? What if one's in the negative axis? Consider e.g. $$\{-n: n \in \mathbb{N} \}$$ This has no lower bound, if one consider lower to mean towards $- \...
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1answer
19 views

Transposed matrix index notation confusion

In an attempt to understand tensors I am reading this document. In page 15 we have $A^{\mu}_{\alpha}a^{\alpha}A^{\mu}_{\beta}b^{\beta}=(A^T)^{\mu}_{\alpha}A^{\mu}_{\beta}a^{\alpha}b^{\beta}$. I can'...
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0answers
25 views

Taylor Series expansion in two variables with one variable fixed

I'm confused by a Taylor series expansion shown by my professor in a course in partial differential equations. After introducing multi-index notation, we went through defining a Taylor series ...
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1answer
59 views

Is there proof of $\frac{dTr(\log(A))}{dA}=A^{-1}$ when A is symmetric using index notation.

So I am a physicist and I encountered the following derivative in my study of the SYK model: $\frac{dTr(\log(A))}{dA}$ where $A$ is a symmetric matrix. I know that Tr$(\log(X))=\log(\det(X))$ and ...
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0answers
16 views

Index notation - notation for the inverse change of basis matrix with a hermitian metric

So, when we have any symmetic bilinear form $g = g_{ij} \epsilon^i \otimes \epsilon^j$, we can write $(A^{-1})^\mu{}_i = A_{\space i}{}^\mu$. This is one of the most beautiful things that index ...
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1answer
28 views

Sum and product notation

I'm working with logic, but I need help with notation. I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs ...
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1answer
10 views

How to write summation of squared divergence terms in index summation notation?

Sorry if this has already been asked before, but it's really difficult to try and explain the problem in words. Anyways, I want to express the following: $$\phi = \left(\frac{\partial u_1}{\partial ...
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1answer
49 views

Vector calculus identities using Einstein index-notation

I have a problem proving these formulas using Einstein index notation. The formulas are: 1) $$\nabla(r^n)= nr^{n-2} \vec{r}$$ 2) $$\nabla \cdot (\nabla g \times \nabla f)=0$$ 3) $$\nabla \times (\...
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3answers
65 views

Tensor equation [closed]

If you have two antisymmetric tensors $A_{\mu \nu}$ and $B_{\mu \nu}$, and for every anti symmetric tensor $\epsilon^{\mu \nu}$, $\epsilon^{\mu \nu} A_{\mu \nu} = \epsilon^{\mu \nu} B_{\mu \nu}$ Is ...
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1answer
14 views

How do I write this multiple index tensor equation using Ricci calculus where the 4-gradient is acting on each tensor?

In the context of special relativity I have this to show that this equation is correct: $$\partial_\mu F^{\mu \nu}=j^\nu$$ To do that I'm trying to take this equation: $$F^{\mu \nu}=\partial^{\mu}A^{\...
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1answer
25 views

Find $\bigcap \mathcal A^c$ and $\bigcup \mathcal A^c$ with the proof.

Let $A_n =\{x \in \Bbb R :-\frac {1}{n} \lt x \lt \frac{1}{n}\}$,$n \in \Bbb N$ and define the indexed family $\mathcal A^c = \{ A_{n}^{c} :n \in \Bbb N \}$. Find $\bigcap \mathcal A^c$ and $\bigcup \...
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2answers
50 views

Derivative of matrix using index notation

In my stats textbook, they define the following function: $\mathbf{f} = \frac{1}{2}(\mathbf{A}\mathbf{x} - \mathbf{b})^2$, where $\mathbf{A}$ is a matrix, $\mathbf{x}, \mathbf{b}$ are just vectors. ...
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1answer
48 views

I wanted to know of book suggestions that can help me overcome my fear of indices

I want to go deeper into General Relativity and Tensor Analysis. However, manipulating the indices always seems to overwhelm me. I wanted to know if there is a good book that covers up that and also ...
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1answer
82 views

Conversion of mixed tensors into mixed tensors and into covariant (or contravariant) ones

I am an undergraduate student of Physics, currently taking a course on Special Relativity, but I am getting too confused with tensors and their indices. My question is: How to convert mixed tensors to ...
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1answer
17 views

Trouble with changing indexing in summation

I have trouble with the following: $$P(t) = \sum_{i=0}^{n-r}c_i^r(tB^{n-r-1}_{i-1}(t)+(1-t)B_i^{n-r-1}(t))$$ $$= \sum_{i=0}^{n-r-1}(tc^r_{i+1}+(1-t)c_i^r)B_i^{n-r-1}(t),$$ where $c_i$ are constants, ...
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2answers
38 views

Calculate the characters from a combination

Given a table with position - 2 character combination pairs like this: 1. aa 2. ab 3. ac 4. ad ... 27. ba 28. bb 29. bc 30. bd Assuming there are unique ...
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1answer
45 views

How to derive the non-torsion-free Bianchi identity by building a canonical torsion-free alternate derivative from the original covariant derivative

I know the Bianchi identity can be derived much more directly and simply (as is apparent in this post). The point here is to follow this alternate path to it as proposed in the paragraph after ...
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2answers
28 views

Order of indices using index notation

I'm trying to solve a problem in Lagrangian mechanics involving index notation. I'm wondering if the expression $$ A_{ij}\dot{q^i}q^j = A_{ji}\dot{q^j}q^i $$ is true. Our professor skipped over this ...
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1answer
35 views

Using diagonality in Einstein notation

Given a diagonal matrix $D$, with diagonal elements given by vector $\mathbf{d}$. Representing this in Einstein notation gives $$ D_{ij} = \delta_{ijk} d_k $$ where $$ \delta_{ijk} = \begin{cases}...
4
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1answer
153 views

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 ...
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2answers
27 views

Changing index of summation

How do you go from $\sum_{n=0}^{\infty} z^{-n-1}$ to $\sum_{n=-\infty}^{-1} z^{n}$ ?? It's really confusing.
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1answer
28 views

Expressing the coordinate dependent and indepent forms of the $(0,1)$ tensor in different coordinate systems

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
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2answers
77 views

Transformation rule in differential geometry

I am reading Walds General Relativity and am looking at Question 8, Chapter 2. In the solutions to this question it states that the metric is determined by the transformation rule $$g_{\alpha\beta}^{'...
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2answers
26 views

Show that $\nabla^{2}\left ( x_{i}x_{j}r^{n} \right ) = 2\delta _{ij}r^{n}+n\left ( n+5 \right )x_{i}x_{j}r^{n-2}$

No matter what I can't seem to arrive at this answer. I've tried $\partial_{i}\left (\partial_{i}\left ( x_{i}x_{j}r^{n} \right ) \right )$ and $\delta _{ij}\partial_{i}\left (\partial_{j}\left ( ...
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1answer
24 views

Ordinary Least Squared derivation using Index Notation

The following function in $\mathbf{x}$ is given: $$f(\mathbf{x}) = \frac{1}{2}\left\lVert \mathbf{Ax - b} \right\rVert_2^2$$ I want to calculate $\frac{\mathrm{d}f(\mathbf{x})}{\mathrm{d}\mathbf{x}}$...
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0answers
30 views

Prove equality with cross products using Einstein notation

Notice that throughout this question I will be using Einstein's summation notation. Let $v(x):\mathbb{R}^3 \to \mathbb{R}^3$ be a vector field and define $$2\mathbb{W}(v) = \nabla v - \nabla v^T$$ ...
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0answers
31 views

Validity of Index Notation

I have the following questions on my biomechanics homework (we're finally getting to multidimensional spaces). However, we haven't gone over it yet and won't have class again until after it's due. I'...
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0answers
206 views

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{...
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0answers
36 views

Show that the laplacian of the curl of A equals the curl of the laplacian of A. $\nabla^2(\nabla\times A) = \nabla \times(\nabla^2A)$

I might have worded the question incorrectly in the title, but I am trying to show that $\nabla^2(\nabla\times \vec A) = \nabla \times(\nabla^2 \vec A)$. I am not sure if there is any physical ...
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0answers
60 views

Tensor and Vector Summation Convention using Index Notation

A second-rank tensor A and a vector v are given by: A= $\begin{bmatrix}1&0&1\\0&2&0\\2&0&3\end{bmatrix}$ v= $\begin{bmatrix}1\\0\\1\end{bmatrix}$ Taking into account the ...
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1answer
45 views

Need help understading prime notation for vector/time series

I have vector/time series with ' symbol at the end: X=(x1,x2,...,xn)' What does it mean? It is derivative of vector/time series?
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3answers
83 views

What is a tensor with two upper or lower indices?

I know very little about tensors. I am trying understand if the following expression would make $$x^a=L^a_{\;b} M^{bc}\hat{x}_c.$$ Einstein summation convention applies. So here is what I think I ...
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2answers
101 views

How does index notation work in Hermitian spaces?

So, I know that in orthogonal spaces (real vector spaces with a symmetric bilinear form) there is a canonical isomorphism bewtween $E$ and $E^*$ induced by the bilinear form $\langle\vec{v}|\vec{w}\...
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2answers
49 views

Matrix and two vectors product

Does anyone know how to represent the product of an $n×n$ matrix and two $n$-vectors in a compact form using $\sum$ and indexes? If it was only a vector then $\sum_{j=1}^{n} A_{i,j}v_j$. But with two ...
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0answers
20 views

Zero Mean Normalized Cross Correlation in Einstein Notation

I try to formulate the Zero-Mean Normalized Cross-Correlation in Einstein Notation. Thus without the Sigma's and with indices. I came up with the following but im not sure if the is correct and ...
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4answers
74 views

What is $10^{40}$ as a number? [closed]

What is $10^{40}$? Every time I google this question I get $1\mathrm{e}\!+\!40$ but I don’t understand this so what is it as a number?
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2answers
48 views

Derivatives Across Summations

So, I took one intro course in Tensor calculus and this problem reminds of that, except I can't quite recall how derivatives work with respect to components, or what those derivatives produce. ...
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1answer
35 views

Strange sum notation

I have the following sum and I would like to generalize the notation but I don't see how to choose i index... $$ -2P(A_{1} \cap A_{2}) - 2P(A_{1} \cap A_{3}) - 2P(A_{2} \cap A_{3}) = -2\sum_{i=...
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0answers
353 views

Examples of Tensor Transformation Law

Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is ...