Index notation Definition and 114 Threads

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. bcrowell

    Notational clash in index notation?

    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...
  2. B

    Help with Index Notation: εlmn detA = εijkAilAjmAkn

    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...
  3. gfd43tg

    Index notation matlab for 2D array

    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...
  4. T

    Index Notation & Dirac Notation

    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...
  5. D

    Matrix notation for Lorentz transformations

    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...
  6. Rococo

    On the index notation used in Lorentz transformations

    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...
  7. N

    Understanding Index Notation and Tensor Operations in Vector Calculus

    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 =...
  8. N

    Index Notation for Rank-2 Tensor with Summation

    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...
  9. binbagsss

    Quick question, index notation, alternating tensor.

    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.
  10. binbagsss

    Index notation/ Tensors, basic algebra questions.

    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...
  11. binbagsss

    Quick question, index notation, alternating tensor.

    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.
  12. S

    Can You Master Basic Index Notation in a Weekend?

    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...
  13. B

    Index notation of matrix tranpose

    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?
  14. M

    Very specific question about index notation

    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...
  15. Markus Hanke

    Is There a Difference Between Covariant and Contravariant Tensor Notations?

    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...
  16. R

    Are Vectors in Index Notation Limited to Using Basis e^l?

    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}}
  17. A

    Relativistic index notation del-operator

    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...
  18. S

    Index Notation Identity for Vector Fields

    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...
  19. T

    Confusing index notation involving grad of w cross r

    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...
  20. A

    Simplifying Index Notation in Vector Calculus

    (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...
  21. S

    Index Notation Help: Solving ∫∂k(gixiεjklxl)dV

    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...
  22. S

    Understanding Division in Index Notation

    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...
  23. S

    Index Notation - Prove the following

    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...
  24. S

    Understanding Index Notation: Allowed Combinations Explained

    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...
  25. X

    Help with conversion to index notation

    Homework Statement Use index notation to calculate the following: Let R = u x (d/du) hint: R_i = epsilon_ijk u_j (d/du_k) R . (u x [T . u]) where T is a traceless (T_ii = 0), symmetric, and constant (i.e, independant of u) second order tensor. Convert your final result to Gibbs'...
  26. C

    GR - Trying to grasp index notation (Levi Civita)

    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] =...
  27. E

    Index Notation: ∂_i 1/r = -x_i/r^3?

    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
  28. B

    Basis vectors and abstract index notation

    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...
  29. J

    Index notation - I never know when to introduce a new symbol?

    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} =...
  30. P

    Is Tab∂cf = Tac∂bf Valid in Tensor Notation?

    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?
  31. L

    Index Notation and Kronecker Delta

    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...
  32. naima

    A question about index notation

    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
  33. Y

    How to denote tetrad in Abstract Index Notation ?

    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?
  34. T

    Vector Calculus: Index Notation

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

    Writing w^2 in Index Notation for Derivation with del X u

    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 Eijk(d^2uk/dxj) where d is the partial derivative, E is the epsilon operator and ijk are suffix's...
  36. Saladsamurai

    Vector Identity Using Index Notation

    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...
  37. G

    Index Notation Help: Understanding Swapping Symbols in AxB=-BxA

    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 =...
  38. L

    How to Prove Vector Calculus Identity Involving Cross Product and Gradient?

    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...
  39. B

    Vector identity proof using index notation

    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) -...
  40. S

    Proving vector identities with index notation (help with the del operator)

    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...
  41. S

    Kronecker delta in index notation

    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...
  42. F

    Help with Beginner Index Notation

    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} =...
  43. N

    Commutator-like notation, index notation

    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...
  44. J

    List of index notation properties ?

    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...
  45. O

    Are These Vector and Matrix Operations Formulated Correctly?

    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...
  46. V

    Proving Relationship: Epsilon-Delta Decomposition for Tensors

    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 =...
  47. V

    Question on Elementary Index Notation

    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...
  48. W

    Understanding Index Notation: Multiplying Vectors & Tensors

    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...
  49. T

    Vector identities in index notation

    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...
  50. H

    Two Occurrences of Same Index in Index Notation?

    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...
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