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. B

    Poisson Brackets / Levi-Civita Expansion

    Hi, I am stumped by how to expand/prove the following identity, \{L_i ,L_j\}=\epsilon_{ijk} L_k I am feeling that my knowledge on how to manipulate the Levi-Civita is not up to scratch. Am i correct in assuming, L_i=\epsilon_{ijk} r_j p_k L_j=\epsilon_{jki} r_k p_i Which...
  2. Z

    Levi-Civita & Jacobi: Meaning & Question

    hey Folks, please have a look at the attached Ex from MTW. does somebody know what is the meaning of the parallel bars in the first levi civita symbol ? Is there a typo in this EX perhaps? I would have expected that on the right hand side one would see the product which is shown in the first...
  3. R

    Levi-Civita Symbol multiplied by itself

    Homework Statement evaluate \epsilon_{ijk}\epsilon_{ijk} where \epsilon is is the antisymetric levi-civita symbol in 3D Homework Equations determinant of deltas = product of levi-civita -> would take ages to write out. The Attempt at a Solution...
  4. A

    Proving vector identity using levi-civita tensor help

    Using the fact that we can write the vector cross-product in the form: (A× B)i =ε ijk Aj Bk , where εijk is the Levi-Civita tensor show that: ∇×( fA) = f ∇× A− A×∇f , where A is a vector function and f a scalar function. Could you please be as descriptive as possible; as I'm not sure...
  5. bcrowell

    Upper- and lower-index Levi-Civita tensor/symbol

    MTW p. 87 defines what they refer to as a Levi-Civita tensor with \epsilon^{\kappa\lambda\mu\nu}=-\epsilon_{\kappa\lambda\mu\nu}. They define its components to have values of -1, 0, and +1 in some arbitrarily chosen Cartesian frame, in which case it won't have those values under a general change...
  6. fluidistic

    Understanding the Properties of Levi-Civita Symbol in Tensor Calculus

    Homework Statement If \epsilon _{ijjk} is the Levi-Civita symbol: 1)Demonstrate that \sum _{i} \epsilon _{ijk} \epsilon _{ilm}=\delta _{jl} \delta _{km} -\delta _{jm} \delta _{kl}. 2)Calculate \sum _{ij} \epsilon _{ijk} \epsilon _{ijl}. 3)Given the matrix M, calculate \sum _{ijk} \sum...
  7. R

    Difference between covariant and contravariant levi-civita tensor?

    The title says it all, basically I'm trying to figure out what the difference is between the two tensors (levi-civita) that are 3rd rank. Do they expand out in matrix form differently?
  8. J

    Kronicker Delta, Levi-Civita, Christoffel and tensors

    Kronicker Delta, Levi-Civita, Christoffel ... and "tensors" For quick reference in grabbing latex equations: http://en.wikipedia.org/wiki/Levi-Civita_symbol http://en.wikipedia.org/wiki/Kronecker_delta http://en.wikipedia.org/wiki/Christoffel_symbols Wiki warns that the Christoffel...
  9. R

    Levi-civita connection assumptions

    In a lot of textbooks on relativity the Levi-Civita connection is derived like this: V=V^ie_i dV=dV^ie_i+V^ide_i dV=\partial_jV^ie_idx^j+V^i \Gamma^{j}_{ir}e_j dx^r which after relabeling indices: dV=(\partial_jV^i+V^k \Gamma^{i}_{kj})e_i dx^j so that the covariant derivative is...
  10. W

    Trying to understand Levi-Civita Symbol and notation

    Hello, I am having a little difficulty understanding what exactly the Levi-Civita symbol is about. In the past I believed that it was equal to 1, -1 and 0, depending on the number of permutations of i,j,k. I had just accepted that to be the extent of it. However, now I am seeing things...
  11. T

    Levi-Civita symbol and Kronecker delta

    Hello everyone, I am stuck when I study Levi-Civita symbol. The question is how to prove \varepsilon_{ijk}\varepsilon_{lmn} = \det \begin{bmatrix} \delta_{il} & \delta_{im}& \delta_{in}\\ \delta_{jl} & \delta_{jm}& \delta_{jn}\\ \delta_{kl} & \delta_{km}& \delta_{kn}\\ \end{bmatrix}...
  12. mnb96

    Expression with levi-civita symbol

    Hello, during a calculation I got the following term: \varepsilon^{i_1 \ldots i_n}\varepsilon_{j_1 \ldots j_n} (a_{i_1}^{j_1}\ldots a_{i_n}^{j_n}) where \varepsilon is the levi-civita symbol and a_i^j are real numbers. Is it possible to simplify that expression?
  13. C

    Relationship between Chern and Levi-Civita Connections on Kahler Manifolds

    So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler. I basically understand the meaning of this statement, but...
  14. M

    Levi-civita and symmetric tensor

    Homework Statement Show that \epsilon_{ijk}a_{ij} = 0 for all k if and only if a_{ij} is symmetric.Homework Equations The Attempt at a Solution The first bit I think is just like the proof that a symmetric tensor multiplied by an antisymmetric tensor is always equal to zero. \epsilon_{ijk} = -...
  15. M

    Evaluate Levi-civita expression

    Homework Statement Evaluate the expression \epsilon_{ijk} \epsilon_{jmn} \epsilon_{nkp} Homework Equations \epsilon_{ijk} \epsilon_{ilj} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} The Attempt at a Solution Let \epsilon_{ijk} = \epsilon_{jki} by permutation of Levi-civita...
  16. 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...
  17. F

    Can you explain the geometric interpretation of the Levi-Civita connection?

    Can anyone help me on this question: Under what relation between vector fields X and Y, the Levi-Civita connection of X with respect to Y, \nabla_{Y}X is 0? Any answers or suggestion will be highly appreciated.
  18. A

    Levi-Civita Tensor: Index Interchange Identity

    Does the following identity hold?: \epsilon_{ijk} a_j b_k = -\epsilon_{ijk} a_k b_j
  19. M

    Torsion, affine development and Levi-Civita connection

    I quote http://en.wikipedia.org/wiki/Torsion_tensor#Affine_developments": I try to apply this to the natural connection on the tangent bundle of M = S2 (or more intuitively, of the surface of the Earth) I mean here natural connection the connection which defines the parallel transport so...
  20. T

    Levi-Civita proofs for divergence of curls, etc

    I've also posted this in the Math forum as it is math as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B}) b) \nabla...
  21. T

    Levi-Civita proofs for divergence of curls, etc

    I've also posted this in the Physics forum as it applies to some physical aspects as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times...
  22. C

    Levi-Civita symbol and Summation

    Okay, this is a derivation from Relativistic Quantum Mechanics but the question is purely mathematical in nature. I presume all you guys are familiar with the Levi-Civita symbol. Well I'll just start the derivation. So we are asked to prove that: [S^2, S_j] =0 Where...
  23. R

    Help deriving Lagrange's Formula with the levi-civita symbol

    Ok, so I'm really at a loss as to how to do this. I can prove the formula by just using determinants, but I don't really know how to do manipulations with the levi-civita symbol. Here's what I have so far: (\vec{B} \times \vec{C})_{i} = \epsilon_{ijk}(B_{j}C_{k})\vec{e_{i}} And I'm...
  24. C

    Proof using Levi-Civita symbol

    Homework Statement Prove \sum_{j,k} \epsilon_{ijk} \epsilon_{ljk} = 2\delta_{il} Homework Equations \epsilon_{ijk} \epsilon_{ljk} = \delta_{il}(\delta_{jj}\delta_{kk} - \delta_{jk}\delta_{kj}) + \delta_{ij}(\delta_{jk}\delta_{kl} - \delta_{jl}\delta_{kk}) + \delta_{ik}(\delta_{jl}\delta_{kk} -...
  25. H

    How to Integrate an Expression with Levi-Civita Tensor in Mathematica?

    Hello. This is more a Mathematica question really, but here it goes anyway. As a consequence of some calculations on high energy physics, I need to integrate an expression that involves a Levi-Civita tensor contracted with four FourVectors (I'm using the FeynCalc package). I'm guessing the...
  26. D

    Levi-civita permutation tensor, and kroneker delta

    Hello, I'm interested in seeing some proof of the identities involving the levi civita permutation tensor and and the kroneker delta. I've discovered the utility and efficiency of these identities in deriving the standard vector calculus identities involving div, grad, and curl, but I'm sort of...
  27. D

    Linear & Vector Algebra: Kronecker delta & Levi-Civita symbol

    Hello all. Happy to have finally found this forum, sorry that it took so long! I'm working through a Vector Algebra tutorial and I am having much difficulty with the concepts of Kronecker deltas and the Levi-Civita symbol. I can't fully grasp either of them intiutively. From what I've been...
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