Contents Foreword: MASS and REU at Penn State University xi Preface xiii Chapter 1. Various Ways of Representing Surfaces and Basic Examples 1 Lecture 1 1 a. First examples 1 b. Equations vs. other methods 4 c. Planar models 8 d. Projective plane and flat torus as factor spaces 9 Lecture 2 11 a. Equations for surfaces and local coordinates 11 b. Other ways of introducing local coordinates 14 c. Parametric representations 16 d. Metrics on surfaces 17 Lecture 3 18 a. More about the M¨ obius strip and projective plane 18 b. A first glance at geodesics 20 c. Parametric representations of curves 22 d. Difficulties with representation by embedding 24 e. Regularity conditions for parametrically defined surfaces 27 v
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