In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: Mathematics is the study of quantity, structure, space and change. ... Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, and predicting physical phenomena through a physical theory. There are three types of theories in physics: mainstream theories, proposed theories and fringe theories. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ...
If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
A general tensor U, components , has a symmetric and antisymmetric part (for each pair of indices), defined as:
(symmetric part),
(antisymmetric part),
and similarly for other indices.
Also,
An antisymmetric tensor A (antisymmetric on indices i and j) has the property that the contraction with symmetric tensor B (symmetric on indices i and j) is identically 0. Proof:
Important antisymmetric tensors in physics include the faraday tensor F in electromagnetism. Electromagnetism is the physics of electromagnetic fields: a field, encompassing all of space, comprised of electrical and magnetic fields. ...
As we have stressed in the Introduction a tensor in TTC is handled as a single (geometric) object, so, in addition to the components, one has to input also the basis elements.
The notation for tensors follows a rather straigthforward generalization of the notation for vectors and 1-forms that we have seen in the previous chapter.
Tensors of type (0, q) which are antisymmetrics on all q indices are called exterior q-forms.
In set theory, the adjective antisymmetric usually refers to an antisymmetric relation.
The term "antisymmetric function" is sometimes used for odd function, although some meanings of antisymmetric are essentiality f(y,x) = -f (x,y).
In linear algebra and theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (usually the exchange of two indices, which becomes the transposition of the matrix) is performed.
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