# What is Index notation: Definition and 114 Discussions

In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program.

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1. ### Index Notation, Identity Matrix

Terms only generate when ##k = i ## ##\left( IA \right)_{ij} = \delta_{ik}A_{kj} = \delta_{ii}A_{ij} = A_{ij}## ##\left( AI \right)_{ij} = A_{ik} \delta_{kj} = A_{ii} \delta_{ij} = A_{ij}## Therefore ##IA = AI## I’m bothered by three repeated indices so I’m questioning my derivation.
2. ### I Consistent matrix index notation when dealing with change of basis

Until now in my studies - matrices were indexed like ##M_{ij}##, where ##i## represents row number and ##j## is the column number. But now I'm studying vectors, dual vectors, contra- and co-variance, change of basis matrices, tensors, etc. - and things are a bit trickier. Let's say I choose to...
3. ### A Wald's Abstract Index Notation: Explaining T^{acde}_b

In the second paragraph on page 25 of Wald's General Relativity he rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sese...
4. ### I Manipulation of 2nd, 3rd & 4th order tensor using Index notation

If I have an equation, let's say, $$\mathbf{A} = \mathbf{B} + \mathbf{C}^{Transpose} \cdot \left( \mathbf{D}^{-1} \mathbf{C} \right),$$ 1.) How would I write using index notation? Here A is a 4th rank tensor B is a 4th rank tensor C is a 3rd rank tensor D is a 2nd rank tensor I wrote it as...
5. ### A Index notation and partial derivative

Hi all, I am having some problems expanding an equation with index notation. The equation is the following: $$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j}$$ I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply. Any hint on this would...
6. ### Solve for x in this index notation problem

ok, by direct substitution i know that either ##x=2## or ##x=4## but i would like to prove this analytically, would it be correct saying, ##xln 2= 2ln x## ##xln_{2}2=2 ln_{2}x## ##x=2 ln_{2}x## ##\frac {1}{2}=\frac { ln_{2}x}{x}## ##ln_{2}x^{1/x}##=##\frac {1}{2}## →##2^{1/2}##=##x^{1/x}##...
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### B Can someone please explain Feynman's index notation?

I found some parts of Vol II, Chapter 25 basically unreadable, because I can't figure out his notation. AFAICT he's using a (+,-,-,-) metric, but these equations don't really make any sense: The first one is fine, and so is the second so long as we switch out ##a_{\mu} b_{\mu}## for ##a_{\mu}...
8. ### Levi-Civita Identity Proof Help (εijk εijl = 2δkl)

I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with: δjj δkl - δjl δkj = δkl - δlk Clearly I did not take the right approach in this proof and have no clue as to how to proceed.
9. ### Vector Cross Product With Its Curl

Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...
10. ### I Confusion about index notation and operations of GR

Hello, I am an undergrad currently trying to understand General Relativity. I am reading Sean Carroll's Spacetime and Geometry and I understand the physics (to a certain degree) but I am having trouble understanding the notation used as well as the ideas for tensors, dual vectors and the...
11. ### A Summation Index Notation in the Transformation Equations

In Chapter 7: Hamilton's Principle, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 258-259, we have the following equations: 1. Upon squaring Equation (7.117), why did the authors in the first term of Equation (7.118) are summing over two...
12. ### I Index notation for inverse Lorentz transform

Hi all, just had a question about tensor/matrix notation with the inverse Lorentz transform. The topic was covered well here, but I’m still having trouble relating this with an equation in Schutz Intro to GR... So I can use the following to get an equation for the inverse...
13. ### Can Einstein Index Notation Help Me Solve Equations in Continuum Mechanics?

Hello I am doing some exercises in continuum mechanics and it is a little bit confusing. I am given the following equations ## A_{ij}= \delta_{ij} +au_{i}v_{j} ## and ## (A_{ij})^{-1} = \delta_{ij} - \frac{au_{i}v_{j}}{1-au_{k}v_{k}}##. If I want to take the product to verify that they give...
14. ### A Meaning of Slot-Naming Index Notation (tensor conversion)

I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it, cannot deduce answer from the text. (text is concise, I think it assumes a bit of familiarity with the knowledge) (a) Convert the following expressions and equations into...
15. ### I Index Notation for Lorentz Transformation

The Lorentz transformation matrix may be written in index form as Λμ ν. The transpose may be written (ΛT)μ ν=Λν μ. I want to apply this to convert the defining relation for a Lorentz transformation η=ΛTηΛ into index form. We have ηρσ=(ΛT)ρ μημνΛν σ The next step to obtain the correct...
16. ### I Exploring Tensor Products: Index Notation

We've been learning about tensor products. In particular, we've been looking at index notation for the tensor products of matrices like these: ## \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)## And ## \left( \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22}...
17. ### I A good reference sheet/manual about Einstein index notation?

I'm not used to Einstein notation and I'm struggling a bit with the more complex examples of it. I got the general gist of it and can follow the basic cases but get sometimes a bit lost when there are a lot of indexes and calculus is involved. All primers I've found online for now only give the...
18. ### I Index Notation, Covector Transform Matrix Rep

Just a couple of quick questions on index notation, may be because of the way I'm thinking as matrix representation: 1) ##V^{u}B_{kl}=B_{kl}V^{u}## , i.e. you are free to switch the order of objects, I had no idea you could do this, and don't really understand for two reasons...

24. ### Index notation: Find F_(μν) given F^(μν)?

Homework Statement Find Fμν, given Fμν= (0 Ex Ey Ez) (-Ex 0 Bz -By) (-Ey -Bz 0 -Bx) (-Ez By -Bx 0) Homework Equations gμν =...
25. ### I How do I convert [D]=[A][ B]T[C] to index notation?

I am having trouble converting [D]=[A][ B]T[C] to index notation. I initially thought it would be Dij=AijBkjCkl but I have doubts that this is correct. Would anyone be able to elaborate on this? Regards
26. ### Vector calculus index notation

Homework Statement prove grad(a.grad(r^-1))= -curl(a cross grad (r^-1)) Homework Equations curl(a x b)= (b dot grad)a - (a dot grad)b +a(div b) - b(div a ) The Attempt at a Solution Im trying to use index notation and get di (aj (grad(r^-1))j) =grad(r^-1) di(aj) +aj(di grad(r^-1))j which is...
27. ### MHB How Do You Calculate the Determinant of a Matrix Using Index Notation?

Making sure I have this right, $|A| = \sum_{i}\sum_{j}\sum_{k} \epsilon_{ijk}a_{1i}a_{2j}a_{3k}$ (for a 3 X 3) and a 4 X 4 would be $|A| = \sum_{i}\sum_{j}\sum_{k} \sum_{l} \epsilon_{ijkl} a_{1i} a_{2j} a_{3k} a_{4l}$ ? Is there any special algebra for these terms? (they could be...
28. ### Values of ##k## for which ##A_{ij}A_{ij} = |\vec a|^2##?

Homework Statement The antisymmetric tensor is constructed from a vector ##\vec a## according to ##A_{ij} = k\varepsilon_{ijk}a_k##. For which values of ##k## is ##A_{ij}A_{ij} = |\vec a|^2##? Homework Equations Identity ##\varepsilon_{ijk}\varepsilon_{klm} =...
29. ### Understanding Index Notation: Simplifying Tensor and Vector Equations

Hello all, long time lurker, first time poster. I don't know if I am posting this in the proper section, but I would like to ask the following: In index notation the term σ_{ik}x_{j}n_{k} is \bf{σx}\cdot\bf{n} or \bf{xσ}\cdot\bf{n}, where ##σ## is a second order tensor and ##x,n## are vectors...
30. ### QFT Index Notation: A Beginner's Guide

Hi. I'm just starting QFT for the first time. I've just finished a course in relativity but I'm confused about the index notation I've found in QFT. Here are 2 examples yi = Σ Mij xj and yj = δij yi . These examples don't seem right after what I have learned in relativity unless the index...
31. ### Proving (A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}: Index Notation Question

I am having trouble showing that ##(A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}## Wouldn't the right side end up having a different outcome? Or can we assume its symmetric?
32. ### Index Notation, multiplying scalar, vector and tensor.

I am confused at why ##V_{i,j}V_{j,k}A_{km,i}## the result will end up being a vector (V is a vector and A is a tensor) What are some general rules when you are multiplying a scalar, vector and tensor?
33. ### Index Notation, taking derivative

Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not...
34. ### Index Notation Help: Solve [a,b,c]^2

1. The problem is: ( a x b )⋅[( b x c ) x ( c x a )] = [a,b,c]^2 = [ a⋅( b x c )]^2 I am supposed to solve this using index notation... and I am having some problems. 2. Homework Equations : I guess I just don't understand the finer points of index notation. Every time I think I am getting...
35. ### Learn the Art of Indexology to Writing Lagrangians with Tensors

I recently read that indexology is the art of writing a Lagrangian by just knowing how many dimensions it has and how to contract tensors. I am very interested in this technique, but I cannot find any reference. Can anyone give me a guidance or a reference?
36. ### Is Λa_b the Same as Λba in Tensor Notation?

Index notation in GR is really confusing ! I'm confused about many things but one thing is the order of index placement , ie. is Λa b the same as Λba ? And if not what is the difference ? Thanks If anyone knows of any books or lecture notes that explain index gymnastics step by step...
37. ### Reversing Indices in Contractions: Can it be Done?

Suppose I have something like \left( \nabla_\mu \nabla_\beta - \nabla_\beta \nabla_\mu \right) V^\mu = R_{\nu \beta} V^\nu Can since all the terms involving ##\mu## on the left and ##\nu## on the right are contractions, can I simply do: \left( \nabla^\mu \nabla_\beta - \nabla_\beta \nabla^\mu...
38. ### Covariant Derivative - where does the minus sign come from?

I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as \nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma \nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma Why is there a minus...
39. ### Master Index Notation: How to Ensure Accuracy in Calculations

I'm not sure if this step on my calculation is correct or not...
40. ### Program that writes tensor equations out

Hi, I'm looking for a program that spits out fully summed index equations. For example T_{ii} in, out comes T_{11}+T_{22}+... and so on, with Einstein summation convention.
41. ### Quick question on Geodesic Equation

Starting with the geodesic equation with non-relativistic approximation: \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma_{00}^{\mu} \left( \frac{dx^0}{d\tau} \right)^2 = 0 I know that ## \Gamma_{\alpha \beta}^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\lambda}} \frac{\partial^2 y^{\lambda}}{\partial...
42. ### Translation from vector calc. notation to index notation

Hi, I want to translate this equation R_{\hat{n}}(\alpha)\vec{x}=\hat{n}(\hat{n}\cdot\vec{x})+\cos\left(\alpha\right)(\hat{n}\times\vec{x})\times\hat{n}+\sin\left(\alpha\right)(\hat{n}\times\vec{x}) to index notation (forget about covariant and contravariant indices). My attempt...
43. ### Index Notation: Understanding LHS = RHS

I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read: Summing them up, \partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} I'm trying to understand how LHS = RHS...
44. ### General Relativity - Index Notation

Homework Statement (a) Find matrix element ##M_{ij}## (b) Show that ##x^j## is an eigenvector of ##M_{ij}## (c) Show any vector orthogonal to ##x^j## is also an eigenvector of ##M_{ij}## Homework EquationsThe Attempt at a Solution Part(a) [/B] \frac{\partial^2 \Phi}{\partial x^i x^j} =...
45. ### Index notation tensors quick question

My text has: ##\frac{\partial x^{a}}{\partial x^{p}}V^{p}-\frac{\partial x^{a}}{\partial x^{r}}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}T^{p}_{qr}V^{r}+\frac{\partial x^{a}}{\partial x^{p}}\frac{\partial }{\partial x^{q}}V^{p}=\frac{\partial x^{a}}{\partial...
46. ### Faraday Tensor and Index Notation

Homework Statement (a) Find faraday tensor in terms of ##\vec E## and ## \vec B ##. (b) Obtain two of maxwell equations using the field relation. Obtain the other two maxwell equations using 4-potentials. (c) Find top row of stress-energy tensor. Show how the b=0 component relates to j...
47. ### Strange Miller Index Notation [00.1]

Do someone knows what this dot "." means? I just know this notation [001]. Thank u Abigale
48. ### Einsteins field equations us what type of index notation?

I know that the metric tensor itself utilizes Einstein summation notation but the field equations have a tensor form so the μ and ν symbols represent tensor information. I'm trying to wrap my head around how Einstein used summation notation to simplify the above field equations but it seems...
49. ### Vector identity proof using index notation

Homework Statement I am trying to prove $$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng...
50. ### MHB Index Notation Proof: Proving $\nabla\cdot\left(\phi\textbf{u}\right)$

Hi Everyone! I'm looking to prove $\nabla\cdot\left(\phi\textbf{u}\right)=\phi\nabla\cdot\textbf{u} + \textbf{u}\cdot\nabla\phi$ in index notation where u is a vector and phi is a scalar field. I'm unsure how to represent phi in index notation. For instance, is the first line like...