## Differential Geometry: A First CourseDifferential Geometry: A First Course is an introduction to the classical theory of space curves and surfaces offered at the under Graduate and Post-Graduate courses in Mathematics. Based on Serret-Frenet formulae, the theory of space curves is developed and concluded with a detailed discussion on fundamental existence theorem. |

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### Contents

Preface | 1 |

The First Fundamental Form and Local Intrinsic | 101 |

Geodesics on a Surface | 160 |

The Second Fundamental Form and Local | 279 |

The Fundamental Equations of Surface Theory | 382 |

Hints and Answers to Exercises | 444 |

455 | |

### Common terms and phrases

asymptotic lines base curve binormal Christoffel symbols circle condition cone coordinates corresponding curves are orthogonal cylinder Definition developable surface differential equation direction coefficients duº Dupin indicatrix dvº edge of regression Example formula function fundamental coefficients fundamental form Gaussian curvature geodesic curvature geodesic equations given curve gives Hence Integrating intersection involute isometric let us find line of striction lines of curvature locus metric Ndvº normal curvature obtain orthogonal trajectories osculating plane parallel parametric curves parametric representation parametric transformation perpendicular position vector principal curvatures principal directions principal normal properties prove r1 and r2 radius right helicoid riºr rix rz ruled surface similar manner sinu ſº space curve spherical curvature straight line surface normal surface of revolution Taking dot product tangent plane unit vector usin values z-axis zero