## Calculus of Variations |

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### Contents

1 Definition of the Space | 2 |

2 The Fundamental Metrio Function and the Definition of Distance | 3 |

3 Hilbert Curves and Generalized Arc Length | 9 |

22 other sections not shown

### Common terms and phrases

accumulation point admissible curve arc length arc of class assume Bolza class C1 compact point set comparison curves Consider const constructed continuous function contradiction contravariant vector convergent coordinate neighborhood coordinate system covariant vector curve connecting curve of class defined definition denoted double point end points equal essentially positive Euler equation Euler vector exist a sequence exists a neighborhood extremal arc extremal field extremals issuing F problem F(xQ field of extremals finite number function is essentially gradient greatest lower bound hence Hilbert curve joining homeomorphic homotopic hyper-surfaces initial values integral interval Jacobian lemma limit point lying manifold metric minimizing arc normal coordinates obtain one-to-one correspondence parameter parameterized points Q positive homogeneous proved relative minimum point second variation shown space straight line sub-arc sufficiently small Take tangent vector unique vanishes vector field y-space