The Boutin Lectures on Introductory Differential Geometry

Slectures by Will Black and Chyuan-Tyng "Roger" Wu

1. Background and Motivation for learning Differential Geometry

© 2013, 2015

The field of differential geometry has existed for several centuries and is quite expansive; yet still a topic of ongoing research. As the name implies, differential geometry involves the use of calculus to study geometry problems. The mathematical language and tools of differential geometry are very applicable to physics and engineering since many objects and interactions can be described (or approximated) using differentiable concepts. We can effectively analyze things ranging from the infinitely small to the infinitely large, as well as everything in between. For example, differential geometry may be used to describe rigid bodies, linkage mechanisms, gears, and even extended to planets, galaxies, and the space-time continuum.

Most differential geometry textbooks and courses assume that the basics of the field are known and thus start off at a fairly high level of abstraction. The purpose of this work is to provide the introduction to the field that is often missing, and give examples of how differential geometry may be applied to physics and engineering. We will be focusing on differential geometry of objects embedded in Euclidean space (i.e. things that remain unchanged under rigid motions), because this description applies to many problems in engineering. As engineers and physicists we often need to analyze the data that we collect, so we will also be focusing on how the tools of differential geometry may be approximated numerically. It is important to note that since we will be using numerical methods, the tools we will discuss will not perform well on very noisy data or high texture objects.

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