Page 40 - Intro to Tensor Calculus

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~
~
~
similar manner it can be demonstrated that for (E 1 , E 2 , E 3 ) a given set of basis vectors, then the reciprocal
basis vectors are determined from the relations
1 1 1
~
~
~
~
~
~
~ 2
~ 1
~ 3
E = E 2 × E 3 , E = E 3 × E 1 , E = E 1 × E 2 ,
V V V
~
~
~
where V = E 1 · (E 2 × E 3 ) 6= 0 is a triple scalar product and represents the volume of the parallelepiped
having the basis vectors for its sides.
~
~ 1 ~ 2 ~ 3
~
~
~
Let (E 1 , E 2 , E 3 )and (E , E , E ) denote a system of reciprocal bases. We can represent any vector A
~
~
~
~
with respect to either of these bases. If we select the basis (E 1 , E 2 , E 3 ) and represent A in the form
~
3 ~
2 ~
1 ~
A = A E 1 + A E 2 + A E 3 , (1.2.1)
~
2
~
3
1
~
~
then the components (A ,A ,A )of A relative to the basis vectors (E 1 , E 2 , E 3 ) are called the contravariant
~
components of A. These components can be determined from the equations
~ ~ 1
~ ~ 2
1
2
~ ~ 3
3
A · E = A , A · E = A , A · E = A .
~
~ 1 ~ 2 ~ 3
Similarly, if we choose the reciprocal basis (E , E , E ) and represent A in the form
~
~ 3
~ 2
~ 1
A = A 1 E + A 2 E + A 3 E , (1.2.2)
~ 1 ~ 2 ~ 3
then the components (A 1 ,A 2 ,A 3 ) relative to the basis (E , E , E ) are called the covariant components of
~
A. These components can be determined from the relations
~ ~
~ ~
~ ~
A · E 1 = A 1 , A · E 2 = A 2 , A · E 3 = A 3 .
The contravariant and covariant components are different ways of representing the same vector with respect
to a set of reciprocal basis vectors. There is a simple relationship between these components which we now
develop. We introduce the notation
~ i ~ j
~
~
E i · E j = g ij = g ji , and E · E = g ij = g ji (1.2.3)
where g ij are called the metric components of the space and g ij are called the conjugate metric components
of the space. We can then write
~ ~ ~ 1 ~ ~ 2 ~ ~ 3 ~
A · E 1 = A 1 (E · E 1 )+ A 2 (E · E 1 )+ A 3 (E · E 1 )= A 1
~ ~ 1 ~ ~ 2 ~ ~ 3 ~ ~
A · E 1 = A (E 1 · E 1 )+ A (E 2 · E 1 )+ A (E 3 · E 1 )= A 1
or
3
1
2
A 1 = A g 11 + A g 12 + A g 13 . (1.2.4)
~ ~
~ ~
In a similar manner, by considering the dot products A · E 2 and A · E 3 one can establish the results
1 2 3 1 2 3
A 2 = A g 21 + A g 22 + A g 23 A 3 = A g 31 + A g 32 + A g 33.
These results can be expressed with the index notation as
k
A i = g ik A . (1.2.6)
~ ~ 1
~ ~ 2
~ ~ 3
Forming the dot products A · E , A · E , A · E it can be verified that
ik
i
A = g A k . (1.2.7)
The equations (1.2.6) and (1.2.7) are relations which exist between the contravariant and covariant compo-
~
~
~
~
~ j
nents of the vector A. Similarly, if for some value j we have E = α E 1 + β E 2 + γ E 3 , then one can show
ij ~
~ j
that E = g E i . This is left as an exercise.