Elementary Topics in Differential Geometry (Undergraduate Texts in Mathematics)

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This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. the calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level.

Elementary Topics in Differential Geometry (Undergraduate Texts in Mathematics) Review

As a math undergrad at Kent State University some twenty-odd years ago, I took a course in differential geometry. This was the text; I still have my copy. (Autographed by the author, in fact; I met him on a visit to his university, where I subsequently attended grad school.)The title of this book states, accurately, that its subject matter is 'elementary topics _in_ differential geometry'. This is one of those 'transition' books that introduces students familiar with Subject A to a more-or-less-systematic smattering of elementary topics in Subject B. Here, Subject A is multivariate calculus and Subject B is, of course, differential geometry.Since that's what this book is for, there are way more numbers and pictures in it than you'll ever see in a modern graduate-level differential geometry text. The idea is to show the student the geometric meaning behind all the advanced calculus and help him/her understand _both_ words in the name 'differential geometry'. In short, much of the motivation here is geometric.I liked it a lot and I am still grateful for its highly accessible introduction to a fascinating field. However, I must also add that its approach is not representative of any graduate-level math course I ever took. Of course this is an undergraduate text and isn't supposed to represent graduate-level coursework. Nevertheless, it _may_ give a student the wrong idea about what to expect in more advanced treatments. (Is there some personal history lurking behind that remark? You guess.)An excellent 'transitional' book, then, and highly recommended to readers who want to connect their knowledge of multivariate calculus to the geometry of Euclidean space. It's also a fine example of an expository work on mathematics that remembers its target audience. However, as other reviewers have commented, it needs some answers to the exercises in order to be really useful for self-study. Help other customers find the most helpful reviews� Was this review helpful to you? Yes No Report abuse | PermalinkComment�Comment

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