Disclamer: This is a student prepared note based on the lecure of Tuesday October 1st.
For a function
defined on
to be an extremum of
, it must be that for any function
defined on
that preserves the endpoints of
(that is,
and
), we have ![{\displaystyle {\frac {d}{d\epsilon }}J(y+\epsilon h)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3a51f32d6674946b826edcd3a5626fa0875fc0)
.
![{\displaystyle {\frac {d}{d\epsilon }}J(y+\epsilon h)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3a51f32d6674946b826edcd3a5626fa0875fc0)
Let
signify F differentiated with respect to its nth variable.
(integrating by parts)
Due to the constraints of
and
,
.
As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that
, or in other terms,
.