Index notation Definition and 114 Threads

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...

Classical Theory of Particles and Fields, by Boris Kosyakov, has the following in appendix A: (I don't own a copy of the book. This came to my attention through this physics.SE question, and it turned out that this part of the book is accessible through Amazon's peephole.) This completely...

Homework Statement Show: εlmn detA = εijkAilAjmAkn Homework Equations not sure The Attempt at a Solution I'm really not sure where to begin. I understand index notation (at least I think I do). Can anyone help me get started? I think that once I have the first step, I'll be...

Homework Statement Homework Equations The Attempt at a Solution Hello, I am having some confusion over the notation used in matlab. I don't really know what they mean A = [1:3; 4:6; 7:9] A = 1 2 3 4 5 6 7 8 9 A(1:2, 1:2) ans = 1 2...

Quantum Mechanics using Index notation. Is it possible to do it? I really don't get the Dirac Notation, and every-time I encounter it, I either avoid the subject, or consult someone who can read it. There doesn't seem to be any worthy explanation about it, and whenever I ask what is the Hilbert...

I'm having some confusion with index notation and how it works with contravariance/covariance. (v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j (v_{new})^i=J^i_{\ j}(v_{old})^j (v_{new})_i=\frac{\partial (x_{old})^j}{\partial (x_{new})^i}(v_{old})_j...

I understand how contravariant 4-vectors transform under a Lorentz transformation, that is: ##x'^μ= \Lambda^\mu~_\nu x^\nu## [1] and how covariant 4-vectors transform: ##x'_\mu=(\lambda^{-1})^\nu~_\mu x_\nu##. [2] Now, I have come across the following relations...

Homework Statement Hi I have a vector v. According to my book, the following is valid: \frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v I disagree with this, because the first term on the LHS I can write as (partial differentiation) \frac{1}{2}\partial_i v_jv_j =...

Homework Statement I have the following rank-2 tensor T = \nabla \cdot \sum_{i}{c_ic_ic_i} I would like to write this using index notation. According to my book it becomes T_{ab} = \partial_y \sum_{i}{c_{ia}c_{ib}c_{iy}} Question: The change \nabla \rightarrow \partial_y and c_i...

Q) I am using index notation to show that ε^{0123}=-1 given that ε_{0123}=1. The soluton is: ε^{0123}=g^{00}g^{11}g^{22}g^{33}ε_{0123}=-ε_{0123} where g_{\alpha\beta} is the metric tensor. I am struggling to understand the last equality. Many thanks for any assistance.

Ok I have T_{ij}=μS_{ij} + λ δ_{ij}δS_{kk}. I am working in R^3. (I am after S in terms of T) . I multiply by δ_{ij} to attain: δ_{ij}T_{ij}=δ_{ij}μS_{ij} + δ_{ij} λ δ_{ij}δT_{kk} => T_{jj}=δ_{jj}λS_{kk}+μS_{jj} * My question is , for the LH term of * I choose T_{jj} rathen than T_{ii}. I...

Q) I am using index notation to show that ε^{0123}=-1 given that ε_{0123}=1. The soluton is: ε^{0123}=g^{00}g^{11}g^{22}g^{33}ε_{0123}=-ε_{0123} where g_{\alpha\beta} is the metric tensor. I am struggling to understand the last equality. Many thanks for any assistance.

A week or two ago we went through index notation in class, however I didn't understand it when the lecturer was going through it thus I need to go through it now. I have this weekend to go through it along with other material. Is it possible to go over basic index notation in this short period...

Zee writes in Einstein Gravity in a nutshell page 186 "let us define the transpose by ##(\Lambda^T)_\sigma^\mu = \Lambda_\sigma^\mu##" and even emphasizes the position of the indexes. Yes, they are not exchanged! This must be a typo, right?

I am reading through this text http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf and am having a bit of trouble with one of the arguments that is put in index notation. Specifically, equation (3.3). I was wondering if anyone could have a look at it and clear up a...

I am just wondering, is there a difference in meaning/definition between the indices of a tensor being right on top of each other A_{\mu }^{\nu } and being "spaced" as in A{^{\nu }}_{\mu } I seem to remember that I once read that there is indeed a difference, but I can't remember what it...

given the vector in the first equation below, does that necessarily imply the third equation, as shown? {{u}_{a}}{{e}^{a}}={{x}_{a}}{{e}^{a}} {{u}_{a}}{{e}^{l}}g_{l}^{a}={{x}_{a}}{{e}^{l}}g_{l}^{a} {{u}_{a}}{{e}^{l}}={{x}_{a}}{{e}^{l}}

Hi, I've been wondering about this forever and I finally decided to ask on the forums. In relativistic index notation (with c= \hbar =1) with the minkowski metric g\mu\nu=diag(1,-1,-1,-1), the 4-vector x^{\mu}=(t,x,y,z)=(x^0,\vec{x}), and with the del operator defined as \partial_{\mu}\equiv...

Homework Statement Simplify the following, where A and B are arbitrary vector fields: f(x) = ∇\bullet[A \times (∇ \times B)] - (∇ \times A)\bullet(∇ \times B) + (A \bullet ∇)(∇ \bullet B) I know that the correct solution is A \bullet ∇2B, according to my professor. However, I can't...

Homework Statement consider the position vector expressed in terms of its cartesian components, r=xiei. Let w=wjej be a fixed vector whose components wj are constants that do not depend on the xi, so that δwj/δxi = 0 Homework Equations I am trying to evaluate ∇((wXr)^2) The...

(r×∇).(r×∇)=r.∇×(r×∇) now in index notation it is written as, =xi∂jxi∂j-xi∂jxj∂i but when I tried to prove it ,it just came out twice.can anyone tell how it is correct(given is the correct form).i really mean that i was getting four terms which gave twice of above after reshuffling...

Homework Statement ∫ ∂k(gixiεjklxl dV Can anyone make sense of this? I know I'll need to apply the chain rule when taking the derivative, but I'm not quite sure how to proceed. Also, this is part of a larger problem where g is a gravity vector existing purely in the -z direction, but I...

Hello everyone, Recently I started to use index notation, but still the division is not clear for me. I'll mention just some simple examples that I'm not sure about: Does a =\frac{1}{b_i} mean that a = \sum_{i=1}^{3}\frac{1}{b_i} or a = 1 / \sum_{i = 1}^{3}b_i ? Similarly, does a_i...

Homework Statement http://imgur.com/gTapO Homework Equations The Attempt at a Solution The first one is easy, just use the fact that δi = δ/δxi and it reduces to the sum from with i=1,2,3 of δxi/δxi = 1 + 1 + 1 I tried to do a similar thing with the second one, also using...

I'm not sure if this is the correct place to ask this question, so please let me know if there is a better place for me to post it. I'm having trouble understanding index notation. I understand the basics, such as in the following examples: (a x b) = εijkajbk εijkεiab = δjaδkbδjbδka...

Homework Statement I'm trying to grasp how the indices are listed when writing out multiple vector products or divergences or gradients, etc. I'm working with 'An Introduction to General Relativity' by Hughston and Tod.Homework Equations A\wedge B = \varepsilon_{ijk}A_{j}B_{k} [A,B,C] =...

Hello world, Index notation is driving me crazy: why on Earth is ∂_i 1/r = -x_i/r^3 ? I would expect it to be -x_i/r^2... Thanks for commenting

First of all, I'd like to say hi to all the peole here on the forum! Now to my question: When reading some general relativity articles, I came upon this strange notation: T^{a}_{b} = C(dt)^{a}(∂_{t})_{b} + D(∂_{t})^{a}(dt)_{b}. Can someone please explain to me what this means? Clearly...

This isn't strictly a homework problem but anyway... I'm reading through a QFT textbook that is using index notation, and sometimes a new index symbol will be introduced during some mathematics and it always throws me off. I'll give a simple example, take the Minkowski metric: g^{\mu\nu} =...

Is Tab\partialcf = Tac\partialbf, where T is a tensor? Seems to me like you should be able to do this: Tab\partialcf =Tab\deltabc\partialbf =Tac\partialbf Maybe I'm using the Kronecker delta incorrectly. Could someone check this for me?

Homework Statement Simplify the following expressions involving the Kronecker delta in N dimensions. Where possible, write the final result without indices. C_{ns}\delta_{rn} Homework Equations The Attempt at a Solution I know Kronecker delta is symmetric but that doesn't seem to help. Is...

Could you please tell me what means B_{[ij} B'_{kl]} where B and B' are bivectors I found it in http://arxiv.org/abs/hep-th/0311162" look at lemma 2.4 thanks

I like Penrose's Abstract Index Notation very much. I am familiar with using Abstract Index Notation to denote Coordinate Basis. But when I try to denote tetrad with Abstract Index Notation, I meet problems. How to denote tetrad in Abstract Index Notation?

Homework Statement [PLAIN]http://img585.imageshack.us/img585/526/indexnotation.jpg The Attempt at a Solution How do I proceed?

Homework Statement I need to write w^2 in suffix notation for a derivation I am doing, where w = del X u Homework Equations (del X u) = w The Attempt at a Solution I think it is E[SIZE="1"]ijk([FONT="Comic Sans MS"]d^2u[SIZE="1"]k/[FONT="Comic Sans MS"]dx[SIZE="1"]j) where d...

Homework Statement I am supposed to verify that \nabla\cdot(\mathbf{u}\times\mathbf{v}) = \mathbf{v}\cdot\nabla\times\mathbf{u} - \mathbf{u}\cdot\nabla\times\mathbf{v}\qquad(1)[/itex] I want to use index notation (and I think I am supposed to, though it does not say to explicitly) to...

Homework Statement I was following along with a proof of AxB=-Bxa it went along the lines of; Let; C=AxB=Ciei D=BxA=Diei for i=1,2,3 and we know Ci=eijkAjBk Di=eijkBjAk we can manipulate B and A to give Bj=BsDeltasj Bk=AmDeltamk so we find; Di=eijkDeltasjDeltamkBsAm =...

Homework Statement Prove the following: (\vec{r}\times\nabla)\cdot(\vec{r}\times\nabla)=r^2\nabla^2-r^2 \frac{\partial^2}{\partial r^2}-2r\frac{\partial}{\partial r} Homework Equations (\hat{e_i}\times\hat{e_j})=\epsilon_{ijk} (\hat{e_i}\cdot\hat{e_j})=\delta_{ij} The Attempt at a...

Homework Statement Using index notation to prove \vec{\nabla}\times\left(\vec{A}\times\vec{B}\right) = \left(\vec{B}\bullet\vec{\nabla}\right)\vec{A} - \left(\vec{A}\bullet\vec{\nabla}\right)\vec{B} + \vec{A}\left(\vec{\nabla}\bullet\vec{B}\right) -...

Homework Statement Prove the vector identity: \left(a\times\nabla\right)\bullet\left(u \times v\right)=\left(a \bullet u \right)\left(\nabla \bullet v \right)+\left(v \bullet \nabla \right)\left(a \bullet u \right)-\left(a \bullet v \right)\left(\nabla \bullet u \right)-\left(u...

Homework Statement what does the expression \delta_{ii} mean? Homework Equations \delta_{ij}=1 if i = j and 0 otherwise The Attempt at a Solution What I'm not sure about is if both indices are in the subscript does this mean i can only use it on a term with a subscript or can it also act on...

Okay, so I'm learning some basic index notation, and I have a few questions... Homework Statement f= scalar field F = vector field so, we are supposed to show that curl(fF) = fcurl(F) + (\nablaf) x F The Attempt at a Solution curl(fF) = [\nabla x (fF))]_{k} =...

Homework Statement There are some equations in the notes on field theory I am reading with notation I have never come across before. Someone told me it was a way of ensuring that the expression was anti-symmetric. I can't find it used the same anywhere else but no explanation is provided...

list of index notation properties ?? Is there a list of index notation properties somewhere on the web ?? I'm just looking for a pdf file that I can reference while manipulating tensors using index notation (and summation convention). I'm not looking for proofs at all, just a quick...

w=∇×u Is this correct? w_i=ε_ijk ∂/(∂x_j ) u_k w and u are the vectors C=(x∙y)z Is this correct? C_i= ∑_i〖(x_i y_j)∙z_i 〗 C, x, y, z are vectors A^T∙A ∙x=A^T∙b Is this correct...

Homework Statement Prove the following relationship: \epsilonpqi\epsilonpqj = 2\deltaij Homework Equations The Attempt at a Solution All I have so far is the decomposition using the epsilon-delta \epsilonpqi\epsilonpqj = \epsilonqip\epsilonpqj \epsilonqip\epsilonpqj =...

I have a question regarding the attached file. How do you get those indicies when you multiply the kronecker deltas with A, B, and C? For instance, C - subscript m remains the same on the left side of the expression, but then becomes C subscript i on the right side. How does this logically...

I have a general question about index notation. For an arbitrary quantity, a, "a" denotes a scalar quantity. "a_i" denotes a vector. "a_ij" denotes a 2nd-order tensor. So, if I have something like "a_i*e_ij*b_j" Would this be like multiplying an nx1 vector, an mxm matrix, and an Lx1 vector...

Homework Statement Prove using index notation that, the x denoting a cross-product. (del x f del g)=del f x del g Homework Equations The Attempt at a Solution dif etc. denote partial derivatives. RHS=eijkdjfdkg LHS-I'm not even quite sure how to write it in index...

Question In index notation, can you have more than two occurances of the same index in the same term? Let me provide and example: Let's say I have a two index tensor, M{\alpha \beta}, and I contract it with itself: M_{\alpha \beta} M^{\alpha \beta} Then let's say I wish to operate on...