## Calculus of Variations: Supplementary Notes and Exercises to 1945-1946 [lecture] Notes at New York University, 1949-1950 |

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### Contents

The shortest connection between two | 6 |

The shortest connection in a plane | 10 |

Problems for which no solutions exist | 13 |

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2-dimensional polar admissible functions admissible surface approximating functions approximation theorem arbitrarily close assumed Bernoulli's boundary curve calculated Calculus of Variations chain rule circle arc clearly constant continuous function contravariant vectors convergence coordinate system covariant vector denoted disk displacement vector end-points Eulcr Euler expression Euler's equation Eulor example exists finite number follows function F function u(x given gradient greatest lower bound implies infinitesimal displacement inner product integral curves integrand function interior point invariancc invariant invariant mathematical Jacobian Laplacian least value lnvariance lower semi-continuous matrix minimum problem n-dimensional polar coordinates necessary condition parameter 9 partial derivatives plane quantity region G satisfying the side sequence shorter great circle shortest connection side condition simplified condition simplified notation solution solved sphere spherical polygonal path straight line segment transform according transformed functional transformed to polar vanishing variable vertical whore x-axis zero