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randombill
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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?
arkajad said:The difference is in their transformation rules. As long as you are within orthonormal systems - you will not see the difference.
arkajad said:General rule:
[tex]\epsilon^{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|\epsilon^{ijk}[/tex]
[tex]\epsilon_{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|^{-1}\epsilon_{ijk}[/tex]
So, for each coordinate system that you need, calculate the Jacobi determinant of the transformation from (or to) the Cartesian one. It may give you a factor in front different from 1.
arkajad said:If you tell me, as an example, at which particular point you have a problem, I will try to help you.
A covariant levi-civita tensor is a mathematical object used in tensor calculus to represent the orientation and relative magnitude of a coordinate system in a given space. It is also known as a covariant permutation symbol or an alternating tensor.
The main difference between a covariant and contravariant levi-civita tensor lies in their transformation properties under a change of coordinates. A covariant levi-civita tensor transforms as a rank-one tensor, while a contravariant levi-civita tensor transforms as a rank-one dual tensor. This means that the components of a covariant levi-civita tensor will change when the coordinate system is transformed, while the components of a contravariant levi-civita tensor will remain the same.
The levi-civita tensor has important physical significance in fields such as fluid mechanics, electromagnetism, and general relativity. It is used to define cross products, calculate determinants, and represent the orientation of a coordinate system in a given space.
In tensor calculus, the levi-civita tensor is used to define the cross product of two vectors, to perform coordinate transformations, and to define the determinant of a matrix. It is also used in various tensor operations, such as contraction and raising/lowering indices.
Yes, the levi-civita tensor is unique up to a scale factor. This means that all levi-civita tensors are equivalent to each other, and can be transformed into one another by multiplying by a constant factor. However, the numerical values of the components of a levi-civita tensor will vary depending on the chosen coordinate system.