# What is Levi-civita: Definition and 77 Discussions

Tullio Levi-Civita, (English: , Italian: [ˈtulljo ˈlɛːvi ˈtʃiːvita]; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics.

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1. ### I Lie dragging vs Fermi-Walker transport along a given vector field

We had a thread long time ago concerning the Lie dragging of a vector field ##X## along a given vector field ##V## compared to the Fermi-Walker transport of ##X## along a curve ##C## through a point ##P## that is the integral curve of the vector field ##V## passing through that point. We said...
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### 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?
3. ### I Action of metric tensor on Levi-Civita symbol

We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then \begin{align} g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr} \end{align} where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor. The...
4. ### I Deriving Contravariant Form of Levi-Civita Tensor

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
5. ### I Help with Levi-Civita manipulation

How do I write the following expression $$\epsilon_{mnk} J_{1n} \partial_i\left[\frac{x_m J_{2i}}{|\vec{x}-\vec{x}'|}\right]$$ back into vectorial form? Einstein summation convention was used here. Context: The above expression was derived from the derivation of torque on a general current...
6. ### A Levi-Civita Connection & Length of Curves in GR

I am studying GR and I have these two following unresolved questions up until now: The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes...
7. ### I Levi-Civita for basis vectors

Hello all, I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors e_1, e_2, e_3 in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows: e_i \times e_j =...

31. ### Sign of Levi-Civita Symbol in spherical coordinates

Hi, I am going through the derivation of an instanton solution (n=1) in Srednicki Chp. 93. Specifically, I went through eqn.s 93.29-93.38. However the sign of the Levi-Civita Symbol is bugging me: It says that in 4D Euclidean space, \epsilon^{1234}=+1 in Cartesian coordinates implies...
32. ### Levi-Civita Connection & Riemannian Geometry for GR

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...
33. ### Levi-Civita symbol for cross products?

Hi. Is using the Levi-Civita symbol to calculate cross-product combos like A x (B x C) allot faster than just using the good old determinant method? I ask because my lecturer in electrodynamics 2 told us it is better, but it seems to me that it's going to cost me time to learn to use this...
34. ### Curl in 5D using levi-civita tensor

i really lost with this. i see two possibilities: (1) something like, \epsilon_{abc}\partial_{a}A_{b}e_{c} with a,b,c between 1 and 5 or (2)like that \epsilon_{abcde}\partial_{a}A_{b} one of the options nears correct? thank's a lot
35. ### The invers of fixed Levi-Civita symbol's element

Homework Statement Hi every one, I am really confused on how to calculate the inverse of this Levi Civita symbole's element \epsilon^{tabc}, I tried to used this equation...
36. ### Need clarification on the product of the metric and Levi-Civita tensor

Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...
37. ### Levi-Civita connection and pseudoRiemannian metric

One of the properties of the unique Levi-Civita connection is that it preserves the metric tensor at each point's tangent space, allowing the definition of invariant intervals between points in the manifold. I'd be interested in clarifying: when the metric preserved by the L-C connection is a...
38. ### Levi-Civita Tensor & Group Theory: Symmetry?

I have been trying to think about the Levi-Civita tensor in the context of Group Theory. Is there a group that it is symmetric to? I'm sorry if this is a double post but I don't think my original identical post submitted correctly. Thanks, Nate
39. ### Pauli matrices and the Levi-Civita tensor : commutation relations

Homework Statement Whats up guys! I've got this question typed up in Word cos I reckon its faster: http://imageshack.com/a/img5/2286/br30.jpg [Broken] Homework Equations I don't know of any The Attempt at a Solution I don't know where to start! can u guys help me out please...
40. ### Proving properties of the Levi-Civita tensor

Homework Statement Hey everyone, So I've got to prove a couple of equations to do with the Levi-Civita tensor. So we've been given: \epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj} We need to prove the following: (1) \epsilon_{ijk}=-\epsilon_{kji} (2)...
41. ### Levi-Civita Identities: Verifying Index Notation Problems

I am having trouble establishing a process to verify various identities for problems in index notation. Description of Problem Verify that \epsilon_{ijk}\epsilon_{iqr}=\delta_{jq}\delta_{kr}-\delta_{kq}\delta_{jr} Attempt at Solution I know that the term is only positive if there is an...
42. ### 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...
43. ### Question involving Levi-Civita symbol

Can someone please explain to me why \epsilon_{ijk}\frac{\partial}{\partial x_i}\frac{\partial A_k}{\partial x_j} = 0 where A is a constant vector field.
44. ### Levi-Civita Tensor product

In many physics literature I have encountered, one of the properties of Levi-Civita tensor is that ε_{ijk}ε_{lmn}is equivalent to a determinant of Kronecker symbols. However this is only taken as a given theorem and is never proved. Is there any source which has proven this property?
45. ### Levi-Civita and Kronecker delta identity, proof with determinants

Homework Statement I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel. What is the author saying in the last line of text? How can we sum the deltas in the upper...
46. ### Levi-Civita Symbol: Understanding Rank 8 Tensor Properties

I am reading Landau and Lifgarbagez's Classical Theory of Fields, 4th edition. In the beginning of page 18, the completely antisymmetric unit tensor is said to be a pseudotensor, because none of it components changes sign when we change the sign of one or three of the coordinates. Then, in...
47. ### Is the Levi-Civita connection unique for a given manifold?

Hi all, To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric. Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics...
48. ### Levi-Civita Identity help

Can somebody show me how \epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k}) Turns in to \epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k} Something about the first \epsilon I'm not seeing here when the terms are moved around.
49. ### How to derive Relation between Levi-civita Density and Kronecker's Delta?

The Relation between Levi-Civita Density and Kroneckers Delta as follows \sum^{3}_{k=1} \epsilon_{mnk} \epsilon_{ijk} = \delta_{mi} \delta_{nj} - \delta_{mj} \delta_{ni} Logically we can satisfy both sides of the expression but Can anyone tell me how to derive this analytically ?
50. ### Product of Two Levi-Civita Symbols in N-dimensions

Dear You, In N-dimensions Levi-Civita symbol is defined as: \begin{align} \varepsilon_{ijkl\dots}= \begin{cases} +1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\ -1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\ 0...