What do you mean by symmetric tensor?
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 the product of a symmetric and antisymmetric tensor?
always zero
Symmetric and antisymmetric tensors The (inner) product of a symmetric and antisymmetric tensor is always zero.
What is an antisymmetric function?
In mathematics, especially linear algebra, and in theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition) is performed.
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.
What is antisymmetric matrix with an example?
An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity. (1) where is the matrix transpose. For example, (2)
How many independent elements are there in a symmetric tensor?
Now, for each of these 6 combinations there are 4(4+1)2=10 independent combinations of α and β, as the tensor is symmetric under the exchange of these two indices. Thus, there are in total 6×10=60 independent components of the tensor.
What is the difference between symmetric and antisymmetric?
As adjectives the difference between symmetric and antisymmetric. is that symmetric is symmetrical while antisymmetric is (set theory) of a relation ”r” on a set ”s, having the property that for any two distinct elements of ”s”, at least one is not related to the other via ”r .
What is symmetric in relations?
A symmetric relation is a type of binary relation. An example is the relation “is equal to”, because if a = b is true then b = a is also true.
What is tensor with example?
A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor. ‘Our loyalties are to the species and the planet.
What is difference between scalar and tensor?
Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor.
What is a totally antisymmetric tensor?
Totally antisymmetric tensors include: Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric) ↑ K.F. Riley; M.P. Hobson; S.J. Bence (2010).
What is an antisymmetric contraction?
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. (antisymmetric part). Similar definitions can be given for other pairs of indices.
What is a tensor in math?
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 U i j k … = U ( i j ) k … + U [ i j ] k … . {\\displaystyle U_ {ijk\\dots }=U_ { (ij)k\\dots }+U_ { [ij]k\\dots }.}
What is an example of antisymmetric?
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.