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Is the Levi-Civita symbol a tensor?
[edit] Is the Levi-Civita symbol a tensor? In the physicist’s conception, a tensor is characterized by its behavior under transformations between bases of a certain underlying linear space. If the most general basis transformations are considered, the answer is no, the Levi-Civita symbol is not a tensor.
How does Levi-Civita symbol transform?
[edit] Transformation properties Except for the factor det(B), the symbol transforms as a covariant tensor under basis transformation. When only transformations with det(B) = 1 are considered, the symbol is a tensor. If det(B) can be ±1 the symbol is a pseudotensor.
What is the rank of Levi-Civita tensor?
rank n
The Levi-Civita tesnor is totally antisymmetric tensor of rank n. The Levi-Civita symbol is also called permutation symbol or antisymmetric symbol. It is named after the Italian mathematician and Physicist Tullio Levi-Civita [1-3].
What is the permutation symbol?
Definition. The permutation symbol, sometimes called the Levi-Civita symbol, can have any number of subscripts. If any two of the subscripts are equal, the symbol evaluates to 0.
Why is Levi-Civita a Pseudotensor?
However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant −1, for example, a reflection in an odd number of dimensions, it should acquire a minus sign if it were a tensor. As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor.
What does the symbol CI stand for which tensor?
Likes(1) Reply(0) Piyush pachauri. Mixed tensor.
Is Levi-Civita a Pseudotensor?
As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix.
Why is Levi Civita a Pseudotensor?
What does n and R stand for in permutations?
permutation
n = total items in the set; r = items taken for the permutation; “!” denotes factorial. The generalized expression of the formula is, “How many ways can you arrange ‘r’ from a set of ‘n’ if the order matters?” A permutation can be calculated by hand as well, where all the possible permutations are written out.
Who is the Levi Civita symbol named after?
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita.
Is the Levi-Civita symbol a pseudotensor or vector?
As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor. As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix.
Is the Levi Civita symbol an orthonormal tensor?
If, however, the underlying space is proper Euclidean and only orthonormal bases are considered, then the answer is yes, the Levi-Civita symbol is a tensor. In order to clarify the answer, it is necessary to consider how the Levi-Civita symbol behaves under basis transformations.
How many dimensions can a Civita symbol be used in?
The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining the general case.