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