Page 123 - Intro to Tensor Calculus

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and
~ j
∂E j j j
~ ~ m ~ m
E i · = − E · E i = − δ i = − . (1.4.43)
∂x k mk mk ik
Then equations (1.4.39) and (1.4.40) become

∂A i j
A i,k = − A j
∂x k ik
∂A i i j
i
A = + A ,
,k k
∂x jk
which is consistent with our earlier definitions from equations (1.4.22) and (1.4.28). Here the first term of
the covariant derivative represents the rate of change of the tensor field as we move along a coordinate curve.
The second term in the covariant derivative represents the change in the local basis vectors as we move
along the coordinate curves. This is the physical interpretation associated with the Christoffel symbols of
the second kind.
We make the observation that the derivatives of the basis vectors in equations (1.4.39) and (1.4.40) are
related since
~ j
~
E i · E = δ j
i
and consequently
~
~ j
∂ ∂E ∂E i
~
~
~ j
~ j
(E i · E )= E i · + · E =0
∂x k ∂x k ∂x k
~
~ j
∂E ∂E i
~
~ j
or E i · = −E ·
∂x k ∂x k
Hence we can express equation (1.4.39) in the form
~
∂A i ∂E i
~ j
A i,k = − A j E · . (1.4.44)
∂x k ∂x k
We write the first equation in (1.4.41) in the form
~
∂E j m ~ i ~ i
= g im E =[jk, i]E (1.4.45)
∂x k jk
and consequently
~
∂E j ~ m i ~ ~ m i m m
· E = E i · E = δ i =
∂x k jk jk jk
(1.4.46)
~
∂E j ~ ~ i ~ i
and · E m =[jk, i]E · E m =[jk, i]δ m =[jk, m].
∂x k
These results also reduce the equations (1.4.40) and (1.4.44) to our previous forms for the covariant deriva-
tives.
j
∂ ~ E
The equations (1.4.41) are representations of the vectors ∂ ~ E i k and k in terms of the basis vectors and
∂x ∂x
reciprocal basis vectors of the space. The covariant derivative relations then take into account how these
vectors change with position and affect changes in the tensor field.
The Christoffel symbols in equations (1.4.46) are symmetric in the indices j and k since
~ ~
r
∂E j ∂ ∂~ r ∂ ∂~ ∂E k
= = = . (1.4.47)
∂x k ∂x k ∂x j ∂x j ∂x k ∂x j