Absolute derivatives along curves

26.21 ABSOLUTE DERIVATIVES ALONG CURVES
this is the analogue of the expression in Cartesian coordinates discussed in
section 26.8.
26.21 Absolute derivatives along curves
In section 26.19 we discussed how to differentiate a general tensor with respect
to the coordinates and introduced the covariant derivative. In this section we
consider the slightly different problem of calculating the derivative of a tensor
along a curve r(t) that is parameterised by some variable t.
Let us begin by considering the derivative of a vector v along the curve. If we
introduce an arbitrary coordinate system ui with basis vectors ei , i = 1, 2, 3, then
we may write v = v i ei and so obtain
dv i
dei
dv
=
ei + v i
dt
dt
dt
k
∂e
dv i
i du
ei + v i k
;
=
dt
∂u dt
here the chain rule has been used to rewrite the last term on the right-hand side.
Using (26.75) to write the derivatives of the basis vectors in terms of Christoffel
symbols, we obtain
dv
dv i
duk
=
ei + Γ j ik v i
ej .
dt
dt
dt
Interchanging the dummy indices i and j in the last term, we may factor out the
basis vector and find
i
dv
duk
dv
=
+ Γijk v j
ei .
dt
dt
dt
The term in parentheses is called the absolute (or intrinsic) derivative of the
components v i along the curve r(t)and is usually denoted by
δv i
duk
duk
dv i
≡
+ Γijk v j
= vi ; k
.
δt
dt
dt
dt
With this notation, we may write
dv
δv i
duk
=
ei = v i ; k
ei .
dt
δt
dt
(26.99)
Using the same method, the absolute derivative of the covariant components
vi of a vector is given by
δvi
duk
≡ vi; k
.
δt
dt
Similarly, the absolute derivatives of the contravariant, mixed and covariant
975