Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.19 COVARIANT DIFFERENTIATION

constant (this term vanishes in Cartesian coordinates). Using (26.75) we write

∂v

∂uj

=

∂vi

∂uj

ei+viΓkijek.

Sinceiandkare dummy indices in the last term on the right-hand side, we may

interchange them to obtain

∂v

∂uj

=

∂vi

∂uj

ei+vkΓikjei=

(

∂vi

∂uj

+vkΓikj

)

ei. (26.86)

The reason for the interchanging the dummy indices, as shown in (26.86), is that

we may now factor outei. The quantity in parentheses is called thecovariant

derivative, for which the standard notation is

vi;j≡

∂vi

∂uj

+Γikjvk, (26.87)

the semicolon subscript denoting covariant differentiation. A similar short-hand

notation also exists for the partial derivatives, a comma being used for these

instead of a semicolon; for example,∂vi/∂uj is denoted byvi,j. In Cartesian

coordinates all the Γikjare zero, and so the covariant derivative reduces to the

simple partial derivative∂vi/∂uj.

Using the short-hand semicolon notation, the derivative of a vector may be

written in the very compact form

∂v

∂uj

=vi;jei

and, by the quotient rule (section 26.7), it is clear that thevi;jare the (mixed)

components of a second-order tensor. This may also be verified directly, using

the transformation properties of∂vi/∂ujand Γikjgiven in (26.84) and (26.78)

respectively.

In general, we may regard thevi;jas the mixed components of a second-

order tensor called the covariant derivative ofvand denoted by∇v. In Cartesian

coordinates, the components of this tensor are just∂vi/∂xj.

Calculatevi;iin cylindrical polar coordinates.

Contracting (26.87) we obtain

vi;i=

∂vi

∂ui

+Γikivk.

Now from (26.83) we have

Γi 1 i=Γ^111 +Γ^212 +Γ^313 =1/ρ,

Γi 2 i=Γ^121 +Γ^222 +Γ^323 =0,

Γi 3 i=Γ^131 +Γ^232 +Γ^333 =0,