The decomposition of into two parts is therefore the same, whether we use or .
It is further of importance that when the system of coordinates is changed, not only is an invariant, but that this is also the case with and separately.
We have therefore
§ 36. For the calculation of we shall suppose to be expressed in the quantities and their derivatives. Therefore (comp. (43))
if we put
Now we can show that the quantities are exactly the quantities defined by (40). To this effect we may use the following considerations.
We know that is a contravariant tensor of the second
As this must hold for every choice of the variations (by which choice the variations are determined too) we must have at each point of the field-figure
- This may be made clear by a reasoning similar to that used in the preceding note. We again suppose and to be zero at the boundary of the domain of integration. Then and vanish too at the boundary, so that
we may therefore conclude that
As this must hold for arbitrarily chosen variations we have the equation