DIFF GEOM: Surfaces, Geodesics, and Curvature      

Coverage of the DIFF GEOM material is combined into a single worksheet that begins with the exploration of the fundamental form.  Thus, this worksheet is repeated as the worksheet for the next section which is on curvature.  The purpose of this worksheet is to provide a coherent introduction to the ideas important in the study of surfaces.  

In particular, we will begin by showing that the study of surfaces is based on the concept of a fundamental form  of a surface, where the fundamental form of a surface is a generalization of the Pythagorean theorem that allows us to measure distances on a surface.  However, among the most important topics in the study of surfaces is the study of the geodesics of a surface, where a geodesics are in some sense the "straightest" curves on a surface.  Geodesics are important in architecture and in theoretical physics, but our immediate concern with geodesics is that they allow us to define the curvature of a surface.  Specifically, geodesics allow us to define normal curvature, which in turn tells us how a surface "bends" in the neighborhood of a given point.  Let's load some packages and get started.

>	restart:with(plots):with(plottools):with(VectorCalculus): SetCoordinates(cartesian[x,y,z]): BasisFormat(true):

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The Fundamental Form of a Surface

Arclength of a Curve on a Surface

DIFF GEOM: Geodesics

DIFF GEOM: Geodesics and Arclength

DIFF GEOM: Calculating Geodesics

DIFF GEOM: Normal and Principal Curvatures  

DIFF GEOM: Mean and Gaussian Curvature

Exercises