Page 95 - Intro to Tensor Calculus

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It is readily verified that the reciprocal basis is

~ 2
~ 3
~ 1
E = γ b e 1 − β b e 2 , E = α b e 2 , E = b e 3 .
~
Consider the problem of representing the vector A = A x b e 1 + A y b e 2 in the contravariant vector form
~ 1 ~ 2 ~ i
A = A E 1 + A E 2 or tensor form A ,i =1, 2.
This vector has the contravariant components

1 ~ ~ 1 2 ~ ~ 2
A = A · E = γA x − βA y and A = A · E = αA y .

Alternatively, this same vector can be represented as the covariant vector

~ 1
~ 2
~
A = A 1 E + A 2 E which has the tensor form A i ,i =1, 2.
The covariant components are found from the relations
~ ~ ~ ~
A 1 = A · E 1 = αA x A 2 = A · E 2 = βA x + γA y .

~ 2
~
~ 1
The physical components of A in the directions E and E are found to be:
~ 1
E A 1 γA x − βA y
~
A · = = p = A(1)
~ 1 ~ 1 2 2
|E | |E | γ + β
~ 2
E A 2
~ αA y
A · = = = A y = A(2).
~ 2
~ 2
|E | |E | α
~
Note that these same results are obtained from the dot product relations using either form of the vector A.
For example, we can write
~ 1 ~ 1
~ 1
~ 2 ~ 1
E A 1 (E · E )+ A 2 (E · E )
~
A · = = A(1)
~ 1
~ 1
|E | |E |
~ 1 ~ 2
~ 2 ~ 2
~ 2
E A 1 (E · E )+ A 2 (E · E )
~
and A · = = A(2).
~ 2 ~ 2
|E | |E |
~
i
In general, the physical components of a vector A in a direction of a unit vector λ is the generalized
dot product in V N . This dot product is an invariant and can be expressed
~
i j
i
i
g ij A λ = A λ i = A i λ = projection of A in direction of λ i