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Difference between revisions of "Du Bois-Reymond lemma"

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340501.png" /> is a continuous function on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340502.png" /> and if for all continuously-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340503.png" /> which vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340505.png" /> the relation
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If $N$ is a continuous function on the segment $[x_1,x_2]$ and if for all continuously-differentiable functions $\eta$ which vanish at $x=x_1$, $x=x_2$ the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340506.png" /></td> </tr></table>
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$$\int\limits_{x_1}^{x_2}\eta'(x)N(x)dx=0$$
  
is valid, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340507.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340508.png" />. Formulated by P. du Bois-Reymond [[#References|[1]]].
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is valid, then $N=\text{const}$ on $[x_1,x_2]$. Formulated by P. du Bois-Reymond [[#References|[1]]].
  
 
The du Bois-Reymond lemma is employed in the calculus of variations to derive the [[Euler equation|Euler equation]] in its integral form. In this proof it is not necessary to assume that the extremum of the functional is attained on a twice-differentiable curve; the assumption of continuous differentiability is sufficient.
 
The du Bois-Reymond lemma is employed in the calculus of variations to derive the [[Euler equation|Euler equation]] in its integral form. In this proof it is not necessary to assume that the extremum of the functional is attained on a twice-differentiable curve; the assumption of continuous differentiability is sufficient.

Latest revision as of 15:14, 27 August 2014

If $N$ is a continuous function on the segment $[x_1,x_2]$ and if for all continuously-differentiable functions $\eta$ which vanish at $x=x_1$, $x=x_2$ the relation

$$\int\limits_{x_1}^{x_2}\eta'(x)N(x)dx=0$$

is valid, then $N=\text{const}$ on $[x_1,x_2]$. Formulated by P. du Bois-Reymond [1].

The du Bois-Reymond lemma is employed in the calculus of variations to derive the Euler equation in its integral form. In this proof it is not necessary to assume that the extremum of the functional is attained on a twice-differentiable curve; the assumption of continuous differentiability is sufficient.

References

[1] P. du Bois-Reymond, "Erläuterungen zu der Anfangsgründen der Variationsrechnung" Math. Ann. , 15 (1879) pp. 283–314
[2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)


Comments

References

[a1] S. [S.V. Fomin] Fomine, "Commande optimale" , MIR (1982) (Translated from Russian)
How to Cite This Entry:
Du Bois-Reymond lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Du_Bois-Reymond_lemma&oldid=33172
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article