Parametrizations. Parametrization of conics. Implicit equations of a curve. Planar curves: implicit, explicit equations and equations in polar coordinates.
Length of a regular curve. Line integral of a scalar-valued function. Arc length parameter (or natural parameter).
Frenet frame. Planes and lines associated with Frenet frame. Projections of a curve on planes.
Curvature and torsion. Radius of curvature. Osculating circle. Frenet Serret formulas.
Significative curves: helices.
Chapter 2: Surfaces
Parametrizations. Parametrizations of quadric surfaces. Implicit and explicit equations. Parametrizations by rotation, translation, etc. Surfaces of revolution.
Tangent plane and Normal line at a regular point. Parametric curves.
First fundamental form. Measure. Surface integral of scalar function.
Second fundamental form. Curvatures of the curves of a surface. Normal curvature.
Asymptotic directions. Asymptotic curves.
Classification of regular points of a surface: elliptic, hyperbolic and parabolic points. Umbilic points and planar points.
Principal curvatures. Gaussian curvature. Mean curvature. Principal directions. Lines of Curvature. Geodesics. Euler formula.
Ruled Surfaces. Directrix and rulings. Parametrization of a ruled surface.
Measure and shape of a ruled surface: first and second fundamental forms. distribution parameter. Properties of the ruled surfaces.
Classification of ruled surfaces: developable surfaces and non-developable surfaces. Properties. Edge of regression. Striction curve.
Bibliography
M.P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, NJ, 1976.
J. Burgos. Curvas y Superficies. Ed. Garcia-Maroto. 2008.
Larson, R., Hostetler, R.P., Edwards, B.H., Calculus II. Houghton Miffin Co., 2005.
Lipschutz, M. M. Schaum's outline of theory and problems of differential geometry, 1969.
J. E. Marsden, A. J. Tromba. Vector Calculus (4th ed.)Freeman, New York 1996.
A. López de la Rica, A. Villa. Geometría diferencial. Clagsa, 1997.
S.L. Salas, E. Hille. Calculus. Ed. Reverté.
R.E. Larson, R.P. Hostetler, BH Edwards. Calculus with Analytic Geometry (6th ed.). DC Heath and Company, New York, NY, 1998.
