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<math> = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx </math> |
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<math> = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx </math> |
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<math> = \int_a^b (F_2 \cdot h + [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h |_a^b</math> (integrating by parts) |
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<math> = \int_a^b (F_2 \cdot h - [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h |_a^b</math> (integrating by parts) |
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Due to the constraints of <math>h(a) = 0</math> and <math>h(b) = 0</math>, <math>F_3 \cdot h |_a^b = 0</math>. |
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Due to the constraints of <math>h(a) = 0</math> and <math>h(b) = 0</math>, <math>F_3 \cdot h |_a^b = 0</math>. |
Revision as of 17:24, 24 October 2012
Disclamer: This is a student prepared note based on the lecure of Tuesday October 2nd.
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,
.