## Theses on the Calculus of Variations Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy, Department of Mathematics |

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### Contents

ANALOGUE OF THE EDGE COilDITIONS | 9 |

Max Coral The EulerLagrange Multiplier Rule | 63 |

Ralph Aubrie Hefner The Condition of Mayer for Discon | 95 |

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### Common terms and phrases

admissible arc admissible variations analogue arguments Bliss boundary value problem calculus of variations characteristic numbers characteristic solutions coefficients conjugate constants corner corollary corresponding curve deduced defined definition denote determinant different from zero differential equations discontinuities double integral end conditions Euler-Lagrange equations exists expression extremal extremaloid finite number follows Freohet func functions of lines funo funotions funotlonals funzioni fy.y given Hence Henoe hypothesis identically zero inequality interval Lagrange problem lemma linearly independent Linoei manifold matrix maximizing arc minimizing arc minimizing surface minimum monopoly problem multiplier rule necessary condition neighborhood normal oaloulus olass one-parameter family ourve parameter partial derivatives problem of Lagrange problem of minimizing proof properties quadratic form relation respect Riesz theorem satisfy the equations satisfying the conditions second variation semi-extremal Sinoe solution of equation Stieltjes integrals Stourgeon sufficient conditions suoh theorem theory tions vanish variables Volterra whioh x^Xg