Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 curver(t) that is parameterised by some variablet.

Let us begin by considering the derivative of a vectorvalong the curve. If we

introduce an arbitrary coordinate systemuiwith basis vectorsei,i=1, 2 ,3, then

we may writev=vieiand so obtain

dv

dt

=

dvi

dt

ei+vi

dei

dt

=

dvi

dt

ei+vi

∂ei

∂uk

duk

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

dt

=

dvi

dt

ei+Γjikvi

duk

dt

ej.

Interchanging the dummy indicesiandjin the last term, we may factor out the

basis vector and find

dv

dt

=

(

dvi

dt

+Γijkvj

duk

dt

)

ei.

The term in parentheses is called theabsolute(orintrinsic) derivative of the

componentsvialong the curver(t)and is usually denoted by

δvi

δt

≡

dvi

dt

+Γijkvj

duk

dt

=vi;k

duk

dt

.

With this notation, we may write

dv

dt

=

δvi

δt

ei=vi;k

duk

dt

ei. (26.99)

Using the same method, the absolute derivative of the covariant components

viof a vector is given by

δvi

δt

≡vi;k

duk

dt

.

Similarly, the absolute derivatives of the contravariant, mixed and covariant