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.
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.
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...
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...
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...
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...
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}##...
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}...
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.
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...
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...
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...
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...
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...
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...
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...
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}...
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...
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...
Homework Statement
Can I, for all purposes, say that Nabla, on index notation, is $$\partial_i e_i$$ and treat it like a vector when calculating curl, divergence or gradient?
For example, saying that $$\nabla \times \vec{V} = \partial_i \hat{e}_i \times V_j \hat{e}_j = \partial_i V_j (\hat{e}_i...
I am following some lecture notes looking at the invariance of Poincare transformation acting on flat space-time with the minkowski metric:
##x'^{u} = \Lambda ^{u}## ##_{a} x^{a} + a^{u} ## [1], where ##a^{u}## is a constant vector and ##\Lambda^{uv}## is such that it leaves the minkowski...
I have just begun reading about Einstein's summation convention and it got me thinking..
Is it possible to represent ∑aibici with index notation? Since we are only restricted to use an index twice at most I don't think it's possible to construct it using the standard tensors (Levi Cevita and...
Homework Statement
I shall be grateful if someone can help me understand this notation:
http://files.engineering.com/getfile.aspx?folder=340bee11-1ba4-49b2-9a31-1a747012d69b&file=1.gif
I know that this notation will finally/should finally give me the below six equations...
Hi PF!
Which way is appropriate for defining del in index notation: ##\nabla \equiv \partial_i()\vec{e_i}## or ##\nabla \equiv \vec{e_i}\partial_i()##. The two cannot be generally equivalent. Quick example.
Let ##\vec{v}## and ##\vec{w}## be vectors. Then $$\nabla \vec{v} \cdot \vec{w} =...
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
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...
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...
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} =...
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...
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...
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?
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?
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...
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...
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?
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...
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...
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...
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.
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...
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...
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} =...
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...
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...
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...
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...