Page 122 - Intro to Tensor Calculus

117

Physical Interpretation of Covariant Differentiation

~
~
~
3
1
2
In a system of generalized coordinates (x ,x ,x ) we can construct the basis vectors (E 1 , E 2 , E 3 ). These
basis vectors change with position. That is, each basis vector is a function of the coordinates at which they
are evaluated. We can emphasize this dependence by writing
r
∂~
~
~
1
2
3
E i = E i (x ,x ,x )= i =1, 2, 3.
∂x i
Associated with these basis vectors we have the reciprocal basis vectors
3
2
1
~ i
~ i
E = E (x ,x ,x ), i =1, 2, 3
~
which are also functions of position. A vector A can be represented in terms of contravariant components as
~ 1 ~ 2 ~ 3 ~ j ~
A = A E 1 + A E 2 + A E 3 = A E j (1.4.35)
or it can be represented in terms of covariant components as
~ 3
~ 2
~ j
~ 1
~
A = A 1 E + A 2 E + A 3 E = A j E . (1.4.36)
~
A change in the vector A is represented as
~
∂A k
~
dA = dx
∂x k
where from equation (1.4.35) we find
~
~
∂A j ∂E j ∂A j
~
= A + E j (1.4.37)
∂x k ∂x k ∂x k
or alternatively from equation (1.4.36) we may write
~
~ j
∂A ∂E ∂A j ~ j
= A j + E . (1.4.38)
∂x k ∂x k ∂x k
We define the covariant derivative of the covariant components as
~
~ j
∂A ∂A i ∂E
~
~
A i,k = · E i = + A j · E i . (1.4.39)
∂x k ∂x k ∂x k
The covariant derivative of the contravariant components are defined by the relation
~
~
∂A ∂A i j ∂E j
~ i
~ i
i
A = · E = + A · E . (1.4.40)
,k k k k
∂x ∂x ∂x
Introduce the notation
~ ~ j
∂E j m ~ ∂E j ~ m
= E m and = − E . (1.4.41)
∂x k jk ∂x k mk
We then have
~ m m i
~ i ∂E j ~ ~ i i
E · k = E m · E = δ m = (1.4.42)
∂x jk jk jk