Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

In order to compare the results obtained here with those given in section

10.10 for orthogonal coordinates, it is necessary to remember that here we are

working with the (in general) non-unit basis vectorsei=∂r/∂uiorei=∇ui.

Thus the components of a vectorv=vieiare not the same as the componentsvˆi

appropriate to the corresponding unit basisˆei. In fact, if the scale factors of the

coordinate system arehi,i=1, 2 ,3, thenvi=vˆi/hi(no summation overi).

As mentioned in section 26.15, for an orthogonal coordinate system with scale

factorshiwe have

gij=

{

h^2 i ifi=j,

0otherwise

and gij=

{

1 /h^2 i ifi=j,

0otherwise,

and so the determinantgof the matrix [gij] is given byg=h^21 h^22 h^23.

Gradient

The gradient of a scalarφis given by

∇φ=φ;iei=

∂φ

∂ui

ei, (26.91)

since the covariant derivative of a scalar is the same as its partial derivative.

Divergence

Replacing the partial derivatives that occur in Cartesian coordinates with covari-

ant derivatives, the divergence of a vector fieldvin a general coordinate system

is given by

∇·v=vi;i=

∂vi

∂ui

+Γikivk.

Using the expression (26.82) for the Christoffel symbol in terms of the metric

tensor, we find

Γiki=^12 gil

(

∂gil

∂uk

+

∂gkl

∂ui

−

∂gki

∂ul

)

=^12 gil

∂gil

∂uk

. (26.92)

The last two terms have cancelled because

gil

∂gkl

∂ui

=gli

∂gki

∂ul

=gil

∂gki

∂ul

,

where in the first equality we have interchanged the dummy indicesiandl,and

in the second equality have used the symmetry of the metric tensor.

We may simplify (26.92) still further by using a result concerning the derivative

of the determinant of a matrix whose elements are functions of the coordinates.