derivatives and we shall determine the variation it undergoes by arbitrarily chosen variations
, these latter being continuous functions of the coordinates. We have evidently
By means of the equations
and
this may be decomposed into two parts
|
(42)
|
namely
|
(43)
|
|
(44)
|
The last equation shows that
|
(45)
|
if the variations
and their first derivatives vanish at the boundary of the domain of integration.
§ 35. Equations of the same form may also be found if
is expressed in one of the two other ways mentioned in § 33. If e.g. we work with the quantities
we shall find
where
and
are directly found from (43) and (44) by replacing
,
,
,
and
etc. by
,
etc. If the variations chosen in the two cases correspond to each other we shall have of course
Moreover we can show that the equalities
exist separately.[1]
- ↑ Suppose that at the boundary of the domain of integration
and
. Then we have also
and
, so that
and from