Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.16 GENERAL COORDINATE TRANSFORMATIONS AND TENSORS

so that the basis vectors in the old and new coordinate systems are related by

ej=

∂u′i

∂uj

e′i. (26.67)

Now, since we can write any arbitrary vectorain terms of either basis as

a=a′

i

e′i=ajej=aj

∂u′

i

∂uj

e′i,

it follows that the contravariant components of a vector must transform as

a′

i

=

∂u′i

∂uj

aj. (26.68)

In fact, we use this relation as the defining property for a set of quantitiesaito

form the contravariant components of a vector.

Find an expression analogous to (26.67) relating the basis vectorseiande′iin the two

coordinate systems. Hence deduce the way in which the covariant components of a vector

change under a coordinate transformation.

If we consider the second set of basis vectors in (26.66),e′i=∇u′i, we have from the chain
rule that

∂uj

∂x

=

∂uj

∂u′i

∂u′i

∂x

and similarly for∂uj/∂yand∂uj/∂z. So the basis vectors in the old and new coordinate
systems are related by

ej=

∂uj

∂u′i

e′i. (26.69)

For any arbitrary vectora,

a=a′ie′

i

=ajej=aj

∂uj

∂u′i

e′

i

and so the covariant components of a vector must transform as

a′i=

∂uj

∂u′i

aj. (26.70)

Analogously to the contravariant case (26.68), we take this result as the defining property
of the covariant components of a vector.

We may compare the transformation laws (26.68) and (26.70) with those for

a first-order Cartesian tensor under a rigid rotation of axes. Let us consider

a rotation of Cartesian axesxithroughanangleθabout the 3-axis to a new

setx′i,i=1, 2 ,3, as given by (26.7) and the inverse transformation (26.8). It is

straightforward to show that

∂xj

∂x′i

=

∂x′i

∂xj

=Lij,