What is symmetric and antisymmetric tensors?
A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. For a general tensor U with components and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: (symmetric part)
What do you mean by tensor density?
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.
Is Levi-Civita tensor antisymmetric?
The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term “symbol” emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.
What is symmetric tensor in physics?
From Wikipedia, the free encyclopedia. In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: for every permutation σ of the symbols {1, 2., r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies.
What is antisymmetric relation example?
An example of antisymmetric is: for a relation “is divisible by” which is the relation for ordered pairs in the set of integers. For relation, R, an ordered pair (x,y) can be found where x and y are whole numbers and x is divisible by y.
Is permutation symbol a tensor?
The symbol can also be interpreted as a tensor, in which case it is called the permutation tensor.
What is permutation tensor?
The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. Recalling the definition of the permutation symbol in terms of a scalar triple product of the Cartesian unit vectors, (1)
How many independent components does an antisymmetric tensor have?
six independent components
The reader should take note that the specific duality we have just described is unique to three-dimensional space; in four dimensions (appropriate for relativity) an antisymmetric rank-2 tensor has six independent components and cannot be expected to provide an alternate representation of a four-vector.
When does a tensor have an antisymmetric sign?
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant .
Can a tensor of rank 2 be decomposed into an anti symmetric pair?
In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries. Totally antisymmetric tensors include:
Which is the sum of its symmetric and antisymmetric parts?
Similar definitions can be given for other pairs of indices. As the term “part” suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in
What is the shorthand notation for anti symmetrization?
A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M ,