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The product of two one- forms, written as defines a linear map which takes two vectors into the reals:
It is therefore a 0/2 tensor. denotes the outer product. It is the formal notation to show how the 0/2 tensor is formed from two one- forms.
Note that this product is non- commutitative since gives a different result [ Assignment 3 ] i.e.
The most general 0/2 tensor is a linear sum of such outer products. So
where are the components of the map f and we have used linearity.
If we take a basis for f as [ 16 components ], then
so we have
Under a Lorentz transformation , the components of f become:
A 0/2 tensor is symmetric if
and anti- symmetric if
It follows that any 0/2 tensor can be uniquely decomposed into a symmetric and anti- symmetric part .
with the symmetric and anti- symmetric parts given by