Page 125 - Intro to Tensor Calculus

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ij
EXAMPLE 1.4-8. (Covariant differentiation) Show that g =0.
,k
k
Solution: Since g ij g jk = δ we take the covariant derivative of this expression and find
i
jk
(g ij g ) ,l = δ k i,l =0
jk jk
g ij g + g ij,l g =0.
,l
jk im
But g ij,l = 0 by Ricci’s theorem and hence g ij g =0. We multiply this expression by g and obtain
,l
m jk
g im g ij g jk = δ g = g mk =0
,l j ,l ,l
which demonstrates that the covariant derivative of the conjugate metric tensor is also zero.

EXAMPLE 1.4-9. (Covariant differentiation) Some additional examples of covariant differentiation
are:
l
(i)(g il A ) ,k = g il A l
,k = A i,k
ij
(ii)(g im g jn A ) ,k = g im g jn A ij
,k = A mn,k

Intrinsic or Absolute Differentiation

i
i
The intrinsic or absolute derivative of a covariant vector A i taken along a curve x = x (t),i =1,... ,N
is defined as the inner product of the covariant derivative with the tangent vector to the curve. The intrinsic
derivative is represented
dx j
δA i
= A i,j
δt dt
j
δA i ∂A i α dx
= − A α (1.4.50)
δt ∂x j ij dt
j
δA i dA i α dx
= − A α .
δt dt ij dt
i
Similarly, the absolute or intrinsic derivative of a contravariant tensor A is represented
δA i i dx j dA i i k dx j
= A ,j = + A .
δt dt dt jk dt
The intrinsic or absolute derivative is used to differentiate sums and products in the same manner as used
in ordinary differentiation. Also if the coordinate system is Cartesian the intrinsic derivative becomes an
ordinary derivative.
The intrinsic derivative of higher order tensors is similarly defined as an inner product of the covariant
derivative with the tangent vector to the given curve. For example,

ij p
δA dx
klm = A ij
δt klm,p dt
ij
is the intrinsic derivative of the fifth order mixed tensor A .
klm