Ebook Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks), by John Oprea
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Differential geometry has a long, wonderful history. It has found relevance in areas ranging from machinery design to the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes together geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors. It is also for students in engineering and the sciences. The mix of ideas offer students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only see geodesics on surfaces, but they will also observe the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.
- Sales Rank: #695164 in Books
- Brand: Brand: The Mathematical Association of America
- Published on: 2007-07-10
- Original language: English
- Number of items: 1
- Dimensions: 9.96" h x 1.34" w x 6.97" l, 2.61 pounds
- Binding: Hardcover
- 510 pages
- Used Book in Good Condition
Review
This is a clearly written book which could serve well as a text for an undergraduate course in differential geometry for either one or two semesters....Overall, this is an impressive book that would be an excellent text for an undergraduate course in differential geometry. --Thomas E. Cecil, Mathematical Reviews
From the Publisher
Designed not just for the math major but for all students of science, this text provides an introduction to the basics of the calculus of variations and optimal control theory as well as differential geometry. It then applies these essential ideas to understand various phenomena, such as soap film formation and particle motion on surfaces.
From the Back Cover
Designed not just for the math major but for all readers of science, this book provides an introduction to the basics of the calculus of variations and optimal control theory as well as differential geometry. It then applies these essential ideas to understand various phenomena, such as soap film formation and particle motion on surfaces.
Most helpful customer reviews
28 of 32 people found the following review helpful.
clearest undergrad differential geometry text around
By A. Ali
This is a very well-written text on modern differential geometry for undergraduates. The content of the book is similar to O'Neill's "Elementary Differential Geometry" (e.g. covariant derivatives, shape operators), but it's easier to read. There are many undergrad texts around -- O'Neill, do Carmo, Pressley -- but this one is the most lucidly written one hands-down.
Afer going through Oprea, one might like to tackle O'Neill's "Elementary Differential Geometry" and Vols 2-4 of Spivak's "Comprehensive Introduction to D.G."
Like O'Neill, Oprea develops surface theory using the shape operator. But Oprea takes shortcuts and doesn't develop the theory in quite the same generality as O'Neill does. For example, Oprea doesn't introduce differential forms and the exterior calculus. As a consequence, Oprea restricts himself to the Serret-Frenet equations whereas O'Neill introduces Cartan's structural equations -- of which Serret-Frenet is simply a special case -- as the method of moving frames in full generality. The structural equations are then used (by O'Neill) in both curve and surface theory.
2 of 2 people found the following review helpful.
that this is an excellent introductory text that fits into the undergraduate curriculum following ...
By Lionel Mordecai
The other reviewers are correct; that this is an excellent introductory text that fits into the undergraduate curriculum following the normal calculus sequence for scientists and engineers.
Sadly, however, its power has been greatly weakened by changes that have been made in MAPLE, an applications program that is used in the book to illustrate geometrical points. Syntactical changes made as MAPLE has evolved from version 10 to version 18 and beyond, have made it impossible to run Oprea' demonstrations. Without these demonstrations, a very interesting and useful book has been rendered less effective.
One can hope that eventual updates and a version three is in the planning process. Perhaps the author can find some interest at MAPLE about speeding up an update of this book.
7 of 8 people found the following review helpful.
Nice introduction and applications of differential geometry
By A Customer
I found this book to be a fine introduction to this subject. I was particularly pleased with the practical examples outlined in the book. Even though I am not extremely proficient with Maple, I found the exercises using this software provided important illustrations of applications.
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