Page 102 - Intro to Tensor Calculus

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An example of a linear form is the dot product relation

~
ϕ(~x)= A · ~x (1.3.67)
~
where A is a constant vector and ~x is an arbitrary vector belonging to the vector space V.
Note that a linear form in ~x can be expressed in terms of the components of the vector ~x and the base
vectors ( b e 1 , b e 2 , b e 3 ) used to represent ~x. To show this, we write the vector ~x in the component form

3
1
2
i
~x = x b e i = x b e 1 + x b e 2 + x b e 3 ,
i
where x ,i =1, 2, 3 are the components of ~x with respect to the basis vectors ( b e 1 , b e 2 , b e 3 ). By the linearity
property of ϕ we can write
1
3
i
2
ϕ(~x)= ϕ(x b e i )= ϕ(x b e 1 + x b e 2 + x b e 3 )
3
2
1
= ϕ(x b e 1 )+ ϕ(x b e 2 )+ ϕ(x b e 3 )
i
3
2
1
= x ϕ( b e 1 )+ x ϕ( b e 2 )+ x ϕ( b e 3 )= x ϕ( b e i )
i
i
Thus we can write ϕ(~x)= x ϕ( b e i ) and by defining the quantity ϕ( b e i )= a i as atensorwe obtain ϕ(~x)= x a i .
~
~
~
Note that if we change basis from ( b e 1 , b e 2 , b e 3 )to(E 1 , E 2 , E 3 ) then the components of ~x also must change.
i
Letting x denote the components of ~x with respect to the new basis, we would have
i ~ i ~ i ~
~x = x E i and ϕ(~x)= ϕ(x E i )= x ϕ(E i ).
~
i
The linear form ϕ defines a new tensor a i = ϕ(E i )so that ϕ(~x)= x a i . Whenever there is a definite relation
~
~
~
between the basis vectors (b e 1 , b e 2 , b e 3 )and (E 1 , E 2 , E 3 ), say,
∂x j
~
E i = i b e j ,
∂x
then there exists a definite relation between the tensors a i and a i . This relation is
∂x j ∂x j ∂x j
~
a i = ϕ(E i )= ϕ( i b e j )= i ϕ( b e j )= i a j .
∂x ∂x ∂x
This is the transformation law for an absolute covariant tensor of rank or order one.
The above idea is now extended to higher order tensors.
y
Definition: ( Bilinear form) A bilinear form in ~x and ~ is a
scalar function ϕ(~x, ~y) with two vector arguments, which satisfies
the linearity properties:
(i) ϕ(~x 1 + ~x 2 ,~y 1 )= ϕ(~x 1 ,~y 1 )+ ϕ(~x 2 ,~y 1 )
(ii) ϕ(~x 1 ,~y 1 + ~y 2 )= ϕ(~x 1 ,~y 1 )+ ϕ(~x 1 ,~y 2 )
(1.3.68)
(iii) ϕ(µ~x 1 ,~y 1 )= µϕ(~x 1 ,~y 1 )
(iv) ϕ(~x 1 ,µ~y 1 )= µϕ(~x 1 ,~y 1 )

for arbitrary vectors ~x 1 ,~x 2 ,~y 1 ,~y 2 in the vector space V and for all
real numbers µ.