What is Einstein notation: Definition and 36 Discussions

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

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  1. G

    A Upper indices and lower indices in Einstein notation

    I have read some text about defining the cross product. It can be defined by both a x b = epsilon_(ijk) a^j b^k e-hat^i and a x b = epsilon^(ijk) a_i b_j e-hat^k why the a and b have opposite indice positions with the epsilon? How to understand that physically?
  2. P

    I Tensor Calculus (Einstein notation)

    Hello, I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
  3. Vanilla Gorilla

    B Beginner Einstein Notation Question On Summation In Regards To Index

    So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question. $$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...
  4. redtree

    I Is my use of Einstein notation correct in this example?

    I am wondering if I am using Einstein notation correctly in the following example. For a matrix ##R## diagonal in ##1##, except for one entry ##-1##, such as ##R = [1,-1,1]##, is it proper to write the following in Einstein notation: ##R_{\alpha} R_{\beta} = \mathbb{1}_{\alpha \beta} ##, such...
  5. T

    I Confusion about Einstein notation

    In Einstein summation convention, the summation occurs for upper indices and its repeated but lower indices. However I have some confusion 1) $${\displaystyle v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}},\ \qquad...
  6. M

    Prove the following identity [Einstein notation]

    Homework Statement [/B] Prove the following identity: \vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A}) Homework Equations Kronecker's...
  7. Theta_84

    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...
  8. S

    I Does ##x_\mu x^\mu## Equal ##(x_\mu)^2## in Einstein Notation?

    Hello! Does ##(x_\mu)^2## actually means ##x_\mu x^\mu## in Einstein notation?
  9. G

    Proper usage of Einstein sum notation

    Homework Statement I'm dealing with some pretty complex derivatives of a kernel function; long story short, there's a lot of summations going on, so I'm trying to write it down using the Einstein notation, for shortness and hopefully reduction of errors (also for the sake of a paper in which I...
  10. L

    Partial derivative of inner product in Einstein Notation

    Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...
  11. M

    Looking for Continuum Mechanics book using Einstein Notation

    I'm taking a course in continuum mechanics this semester and the instructor is using a set of notes to teach out of it, problem is, I don't really like them. Can anyone recommend an engineering/applied physics oriented introductory continuum mechanics textbook that uses the Einstein summation...
  12. R

    Einstein notation and the permutation symbol

    Homework Statement This is my first exposure to Einstein notation and I'm not sure if I'm understanding it entirely. Also I added this class after my instructor had already lectured about the topic and largely had to teach myself, so I ask for your patience in advance... The question is...
  13. D

    Learn Einstein Notation with Examples and Notes

    Hi. Currently I'm taking an advanced particle physics course, and apparently Einstein notation takes up a lot in this course. Unfortunately for me, and several others in this course, we have never had anything with this kind of notation before. And pretty much from day one, we were put right...
  14. mnb96

    Problem with Einstein notation and Levi-Civita symbol

    Hello, I consider the permutations \sigma_i, where i\in \{1,\ldots,n\}, of the following kind: \sigma_i is obtained by choosing the i-th element from (1,..,n) and by shifting it to the first position; for instance \sigma_3 = (3,1,2,\ldots,n). The parity of \sigma_i is clearly (-1)^{i-1}. For...
  15. S

    Einstein Notation for Stored Energy Function

    Dear all, I encounter the following formula (for stain energy function) a lot in physics literature: W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij} where all indices ranges from 1 to 3, both \epsilon and \sigma are 3x3 matrices. My question is what...
  16. S

    Electromagnetic wave equation in Einstein Notation

    Hey! How to transform the equation \bigtriangleup\vec E=\operatorname{div}(\operatorname{grad}(\vec E))=\epsilon_0\cdot\mu_0\cdot\frac{\partial^2\vec E}{\partial t^2} in Einstein Notation? Thank you all for your help!
  17. U

    Einstein Notation - Not sure if my answers are right

    Homework Statement Questions 11 and 12 specifically.. Homework Equations The Attempt at a Solution 11(a) 11(b) 12(a) 12(b) 12(c) 12(d) (I did the last part of 12(d) by normal vector methods and got 2a instead...which is the correct answer)
  18. F

    Einstein notation question

    While taking notes in class, I was trying to write the moment of inertia tensor in Einstein notation as Iij instead of a 3x3 matrix, but when we diagonalized the matrix with diagonal elements Ii, I was confused on how to write it. Ii doesn't work because that means that treats it as a vector...
  19. S

    Simple GR Einstein Notation Question

    In reviewing some basic GR (just to keep my old brain sharp), i was looking at the Einstein notation cinvention and was a bit confused. I see how you do the dot product of say: ei.ej = δij (i.e. 1 or 0) But then the book I'm reading talks about ei.ej or ei.ej. Isn't that just the...
  20. G

    Finding Einstein notation version of a given flow equation

    Homework Statement This is from lecture, not a homework problem per se. But I need assistance. The problem was to write this form of a flow equation in Einstein's notation: Homework Equations \frac{\partial }{ \partial x_1}(K_1 \frac{\partial h}{\partial x_1}) + \frac{\partial }{...
  21. andrewkirk

    Solving Einstein Notation: Summing Subscripts and Superscripts

    \vec{v} = v^i\vec{e}_i = g(\vec{v},\vec{e}_i)\vec{e}_i The last bit is a sum over i but will need a Ʃ because the Einstein rule only applies to matched superscripts and subscripts and here bot the i are subscripts. Even if I write out the metric in the basis it doesn't work...
  22. I

    Determinat formula in Einstein notation

    Hi all, I've been looking around at formulae for determinants (using them for tensor densities) and I just want to clarify that the expression below is correct (i.e. formulae are correct): |M| = \sum^n_{a_1,a_2, \ldots ,a_n = 1} \epsilon_{a_1a_2 \ldots a_n} M_{1a_1}M_{2a_2} \ldots M_{na_n}...
  23. D

    Partial Differentiation with Einstein Notation

    Hi there, I'm writing a research paper and have hit a roadblock, (wikipedia did not help) and one of my collaborators sent me an e-mail that I do not understand. I am attempting to find when the following functional is stationary: T = \int\limits_{\lambda_{1}}^{\lambda_{2}}...
  24. J

    What is Einstein Notation for Curl and Divergence?

    Anybody know Einstein notation for divergence and curl? What I would like to do is give each of these formulas in three forms, and then ask a fairly simple question; What is the Einstein notation for each of these formulas? The unit vectors, in matrix notation...
  25. M

    How can I understand the Einstein summation convention for vector algebra?

    Hi, I am just starting to learn vector algebra with Grad, Div, Curl etc and have in passing come across Einstein notation which seems to make things much more concise. The problem I have is in Finding Div(rn r) where r =xi + yj + zk. The unbold r is the magnitude of r. I have used...
  26. M

    Determinant formula with Einstein notation proof

    Hello, I am supposed to prove that the determinant of a second order tensor (a matrix) is equal to the following: det[A] = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{pi} A_{qj} A_{rk} anyone have any idea how i would go about this? any method is welcome where the determinant of the...
  27. S

    Understanding Einstein Notation

    I thought that when you used a roman letter such as v that you started at 1 instead of 0. For instance if you had: A^v C_{\mu v} Wouldn't that just be: A^1C_{\mu 1} + A^2C_{\mu 2} + A^3C_{\mu 3} ? (this is one of the problems with a solution from Schutz's book and the solution starts...
  28. mnb96

    Einstein notation question

    Hi! I want to write the following expression in Einstein notation: (a_1+\ldots+a_n)(b_1+\ldots+b_n) I tried it by introducing the term I^{ij}=1 and writing: a_ib_jI^{ij} Are you aware of another, more convenient form for this?
  29. G

    Einstein Notation: Xj to Xi Transformation Equations

    From what I'm reading, Xj = aijXi (X is a vector and the subscript for a is i)...though I'm not sure where the aijXi came from. Would this by any chance be a relation for the transformation equations from Xj to Xi?
  30. B

    Inverse of tensor using Einstein notation

    If I have a (1,1) tensor, eg a Lorentz transform, how do I write its inverse? For example: x'^\mu=\Lambda^\mu_\nu x^\nu Would I multiply on the left by: (\Lambda^{-1})^\nu_\mu? It seems to make sense, but I'm not 100% sure. I'd prefer to not use anything from matrix multiplication...
  31. E

    Understanding Einstein Notation for Derivatives of Fields

    Homework Statement Is \partial ^ {\nu} F_{\mu \nu} + m^2 A_{\mu} the same expression as \partial _{\nu} F^{\mu \nu} + m^2 A^{\mu} ? What form of the metric do I need to hit them with to show that? Also is \partial ^{\nu} \partial_{\mu} A_{\nu} = \partial_{\mu} \partial ^{\nu} A_{\nu}...
  32. E

    Einstein Notation: How to Go from 16 Terms to 4 Terms

    Homework Statement I am a bit new to Einstein notation. Why is this true: \eta_{\mu \nu} \frac{d \delta x ^ {\mu}}{d \tau} \frac{d x ^ {\nu}}{d \tau} = \frac{d \delta x ^ {\mu}}{d \tau} \frac{d x _ {\nu}}{d \tau} How can you go from 16 terms on the LHS to just 4 terms on the RHS. If it is...
  33. T

    Please me to understand Einstein notation and permutations

    Hey, Can anyone help me to understand einstein notation and permutations? I have a book, but it's not very clear. I really don't understand how you can write A_ij = e_ijk a_k out as a matrix? To start with I understand that a matrix can be represented as A_ij where i is the row and j is the...
  34. quasar987

    Is Einstein Notation Used Consistently in Tensor Analysis on R³?

    I'm reading a text on tensor analysis (on R³), and I don't understand the following exemple... P=\frac{1}{2}(a_{ij}+a_{ji})x_ix_j=\frac{1}{2}(a_{ij}x_ix_j+a_{ij}x_jx_i)=a_{ij}x_ix_j To pass from the second to the last equality, he commuted the second pair of x_jx_i into x_ix_j. But he...
  35. A

    Quick question on Einstein Notation

    Hello all, I have a quick question on Einstein notation. I'll write the tensors as a capital letter and the covariant indices as lower case letters (and not use anything that has contravariant indices). I'll also use != for not equal or not congruent to. In Schaum's outline of tensor...