Contravariant tensor of the second rank:-
If n 2 quantities A q s in a coordinate system (x 1,x 2,........x n)are related to n 2 other quantities Aˉpr in another coordinate system(x1ˉ,x2ˉ ,............xnˉ) by the transformation equations
Aˉ pr= s=1∑n q=1∑n ∂xq∂xpˉ ∂xs∂xrˉ A q s p,r=1,2,..........,n
or, by our conventions,
Aˉ pr=∂xq∂xpˉ ∂xs∂xrˉ
they are called components of a contravariant tensor of the second rank (or of rank two).
Covariant tensor of the second rank :-
If n 2 quantities A q s in a coordinate system (x 1,x 2,.......x n) are related to n 2 other quantities Aˉ pr in another coordinate system (x1ˉ,.........,xnˉ ) by the transformation equations
Aˉ pr= s=1∑n q=1∑n ∂xpˉ∂xq ∂xrˉ∂xs A q s p,r=1,2,........,n
or, by our conventions,
Aˉ pr=∂xpˉ∂xq ∂xrˉ∂xs
they are called components of a covariant tensor of the second rank.
Show that the Kronecker delta function is a mixed tensor of rank 2
The Kronecker delta or Kronecker tensor, written, δqp and defined by
δqp ={01if p=/qif p=q
is a mixed tensor of rank 2,justify the notation used.
Proof:-
we must prove that
δkjˉ =∂up∂ujˉ ∂ukˉ∂uq δqp
By definition
δqpˉ=δqp= {01if p=/qif p=q
Coordinates ujˉ are functions of coordinates u p which are in turn functions of coordinates ukˉ .Then,by chain rule,
∂up∂ujˉ∂ukˉ∂up=∂ukˉ∂ujˉ=δkj
Now we must prove that the right side of the transformation equations of the mixed tensor above equals δkjˉ .It does:
∂up∂ujˉ∂ukˉ∂uqδqp =∂up∂ujˉ∂ukˉ∂up =∂ukˉ∂ujˉ=δkj=δkjˉ
Rank of Scalar tensor:-
In fact tensors are merely a generalization of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor. The rank (or order) of a tensor is defined by the number of directions (and hence the dimensionality of the array) required to describe it.
A tensor is a mathematical representation of a scalar (tensor of rank zero), a vector(rank 1)
scalar=rank 0->magnitude ,no direction.
vector=rank 1->magnitude and direction.
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