![]() To my relief, the elegant treatment of Archimedes’ theorem, which does not depend on doing any detailed spherical geometry, has been preserved. A section on spherical geometry has also been added, which the author suggests should only be covered if the instructor is aiming for hyperbolic geometry as the final topic. This is something I could do without, but as long as it’s segregated in a chapter of its own, I can safely ignore it. The largest addition to the book is the chapter on hyperbolic geometry. Recognizing that instructors can have different things they want to get to in the course, the author has organized the book so that the first ten chapters contain the core material, while the final three chapters present a choice of “culminating result” one could aim for: hyperbolic geometry, minimal surfaces, and the Gauss-Bonnet theorem. But parallel transport is an important idea, and it’s the way most physicists think about curvature, so students should be exposed to it. The book still approaches curvature in an elementary way, without introducing connections. The other positive change is the inclusion of a discussion of parallel transport. In addition, this way of thinking is just easier! At least in the U.S., every student taking this course is likely to have taken a linear algebra course and encountered these ideas, and every chance to reinforce and use them should be seized. ![]() In this version, the tangent plane is treated as a vector space, and we can therefore talk about operators and quadratic forms on it. ![]() More generally, in the first edition there was no use of the idea of the matrix of a linear transformation with respect to a chosen basis, making it harder to say certain things. In the first edition, the Weingarten operator was never defined, with the result that a certain matrix played a very important - but entirely unexplained - role. The change that pleases me the most is that the new edition makes a little more use of linear algebra. In the light of that, I am very happy to report that the new edition of Pressley’s Elementary Differential Geometry is an even better book than the first edition, which I reviewed some time ago. There is always a temptation, as well, to add stuff - sometimes lots of stuff - requested by users of the book, all of whom, it seems, would like their pet topic to be included. It’s all too common for authors to “improve” a book in ways that make it much less interesting and much more like all the other textbooks, thereby losing most of what made the first edition distinctive and interesting. When a new edition of a book I know and love is announced, I get a little nervous.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |