What is Einstein summation: Definition and 28 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. E

    I From Einstein Summation to Matrix Notation: Why?

    I know that if ##\eta_{\alpha'\beta'}=\Lambda^\mu_{\alpha'} \Lambda^\nu_{\beta'} \eta_{\alpha\beta}## then the matrix equation is $$ (\eta) = (\Lambda)^T\eta\Lambda $$ I have painstakingly verified that this is indeed true, but I am not sure why, and what the rules are (e.g. the ##(\eta)## is in...
  2. D

    I Einstein summation convention in QM

    Hi For an operator A we have Aψn = anψn ; the matrix elements of the operator A are given by Amn= anδmn My question is : is this an abuse of Einstein summation convention or is that convention not used in QM ? Thanks
  3. D

    I Einstein summation convention confusion

    Hi If i have a vector r = ( x1 , x2 , x3) then i can write r2 as xixi where the i is summed over because it occurs twice. Now is xixi the same as xi2 ? I have come across an example where they are used as equivalent but i am confused because xi2 seems to be the square of just one component of r...
  4. Athenian

    Einstein Summation Convention Question 2

    Below is my attempted solution: $$\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$$ $$\Rightarrow (\delta_{ik} \, \delta_{jm} - \delta_{im} \, \delta_{jk}) \epsilon_{ijm} \, a_k$$ $$\Rightarrow \delta_{ik} \, \delta_{jm} \, \epsilon_{ijm} \, a_k - \delta_{im} \, \delta_{jk}...
  5. Athenian

    How Can I Simplify and Solve the Einstein Summation Convention Problem?

    Attempted Solution: $$a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}$$ $$a_i\, b_j\, c_k\, (\delta_{i3} \, \delta_{jk} - \, \delta_{ik}\, \delta_{j3})$$ Beyond this, though, I am quite lost. I know I am very close to the answer, but seeing this many terms can become fairly...
  6. Hiero

    B Einstein summation notation, ambiguity?

    If I see ##f(x_ie_i)## I assume it means ##f(\Sigma x_ie_i)## (summing in the domain of f) but what if I instead wanted to write ##\Sigma f(x_ie_i)## (summing in the range)? Is there a way to distinguish between these in Einstein’s summation notation?
  7. M

    I Christoffel symbol and Einstein summation convention

    Homework Statement I know that by definition Γijkei=∂ej/∂xk implies that Γmjk=em ⋅ ∂ej/∂xk (e are basis vectors, xk is component of basis vector). Can I write it in the following form? Γjjk=ej ⋅ ∂ej/∂xk Why or why not? Homework EquationsThe Attempt at a Solution
  8. M

    How to Write the Inverse of a Matrix Using Einstein Summation Notation?

    Homework Statement I am unsure as to how to write the dot product in terms of the summation notation? May you please explain? Homework EquationsThe Attempt at a Solution
  9. P

    Einstein summation convention and rewriting as a matrix

    Homework Statement The question asks us to write down the matrix represented by the following summation. 2. Homework Equations The question summation... $$\sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k}$$ The Attempt at a Solution $$ \sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k} =...
  10. Raptor112

    Einstein Summation: Swapping Dummies i & j

    Homework Statement My question is regarding a single step in a solution to a given problem. The step begins at: ##\large \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{y^p} \frac{\partial x^j}{\partial y^q} - \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{\partial...
  11. D

    Einstein summation notation

    I have been looking through some notes on fermion wavefunction operators and noticed some summations involving indexes repeated 3 times.I know this is not allowed when using the Einstein summation convention. So my question is : is the Einstein convention not used in Quantum mechanics ? and do...
  12. P

    Division with Einstein summation convention

    Homework Statement I have the following equation Aab= c ua ub Where Aab is a rank 2 tensor and ua is a vector and c is a scalar and a,b = {0,1,2,3}. I know both Aab , ua and ua I want to find c explicitly but I don't know how to interpret or calculate c = Aab / ( ua ub ) Does anyone...
  13. N

    Divergence of a rank-2 tensor in Einstein summation

    Homework Statement Hi When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} +...
  14. M

    Einstein summation notation for magnetic dipole field

    I can do this derivation the old fashioned way, but am having trouble doing it with einstein summation notation. Since \vec{B}=\nabla \times \vec{a} \vec{B}=\mu_{0}/4\pi (\nabla \times (m \times r)r^{-3})) 4\pi \vec{B}/\mu_{0}=\epsilon_{ijk} \nabla_{j}(\epsilon_{klm} m_{l} r_{m} r^{-3})...
  15. M

    How Do You Use Einstein Summation to Prove Vector Calculus Identities?

    prove the identity $$\nabla\times(f\cdot\vec{v})=(\nabla f) \times \vec{v} + f \cdot \nabla \times \vec{v}$$ I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with einstein summation notation, but am having some trouble since I am...
  16. H

    Bras and kets vs. Einstein summation convention

    Greetings, This is just an opinion question about notations. Having learned the basics of bra-ket notation and using the ESC, as far as I can tell, ESC is just plain better, at least when dealing with finite bases. Using bras and kets, you can represent and manipulate states using...
  17. J

    Determining the commutation relation of operators - Einstein summation notation

    Determining the commutation relation of operators -- Einstein summation notation Homework Statement Determine the commutator [L_i, C_j] . Homework Equations L_i = \epsilon_{ijk}r_j p_k C_i = \epsilon_{ijk}A_j B_k [L_i, A_j] = i \hbar \epsilon_{ijk} A_k [L_i, B_j] = i \hbar...
  18. C

    Regarding Einstein Summation Convention

    So, I realize the basic theory behind Einstein Summation Convention is that any repeated set of indices implicitly indicates a sum over those indices. However, what if an index is repeated three times? For example, my mathematics professor posted this problem: εijkajaj = ? As you can...
  19. L

    Confusion over Einstein summation convention and metric tensors.

    My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...
  20. J

    Vector field identity derivation using Einstein summation and kronecker delta.

    Homework Statement Let \vec{A}(\vec{r})and \vec{B}(\vec{r}) be vector fields. Show that Homework Equations \vec{\nabla}\bullet(\vec{A}\vec{B})=(\vec{A}\bullet\vec{\nabla})\vec{B}+\vec{B}(\vec{\nabla}\bullet\vec{A}) This is EXACTLY how it is written in Ch 3 Problem 2 of Schwinger...
  21. B

    What Do These Einstein Summation Convention Expressions Represent?

    Homework Statement Ok so I'm meant to answer: To what scalar or vector quantities do the following expressions in suffix notation correspond? (expand and sum where possible). 1) aibjci 2) aibjcjdi 3) dijaiaj 4) dijdij 5) eijkaibk 6)eijkdij Homework Equations The...
  22. I

    Why Do Subscripts and Superscripts Vary in Einstein Summation Notation?

    hi i am just reading some notes on tesor analysis and in the notes itself while representing vectors in terms of basis using einstein summation notation the author switches between subsripts and superscripts at times. are there any different in these notation. if so what are they and when should...
  23. J

    How to Use Einstein's Summation Convention for Gradient Calculations?

    Homework Statement Basically need to use einstein's summation convention to find the grad of (mod r)^n and a.r where a is a vector and r = (x,y,z) Homework Equations The Attempt at a Solution Not sure where to begin really.. :S grad (mod r)^n= (d/dx, d/dy, d/dz) of root (X1^2...
  24. T

    Einstein summation convention proof

    Homework Statement Using the Einstein summation convention, prove: A\bulletB\timesC = C\bulletA\timesB Homework Equations The Attempt at a Solution I tried to follow an example from my notes, but I don't entirely understand it. Would it be possible to find out if what I've...
  25. tony873004

    Einstein Summation Convention, Levi-Civita, and Kronecker delta

    Homework Statement Evaluate the following sums, implied according to the Einstein Summation Convention. \begin{array}{l} \delta _{ii} = \\ \varepsilon _{12j} \delta _{j3} = \\ \varepsilon _{12k} \delta _{1k} = \\ \varepsilon _{1jj} = \\ \end{array} The Attempt at a...
  26. R

    Einstein Summation Convention / Lorentz Boost

    Einstein Summation Convention / Lorentz "Boost" Homework Statement I'm struggling to understand the Einstein Summation Convention - it's the first time I've used it. Would someone be able to explain it in the following context? Lorentz transformations and rotations can be expressed in...
  27. Y

    How to deal with the index in Einstein summation?

    Given U^k_i, the components of U is a delta function i.e for i=k U^i_k =1, to prove it is invariant under Lorentz transformation~~ I don't know how to express it in Einstein summation notation, I am very confused with the upper-lower index, is it right to write the transformation in this...
  28. K

    Different between integrals and Einstein summation?

    From http://mathworld.wolfram.com i see that the integral notation was "the symbol was invented by Leibniz and chosen to be a stylized script "S" to stand for 'summation'. " So from that i figure integrals are just summations. So what's the difference from Einstein Summation, where "repeated...