Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks) [ILLUSTRATED] (Hardcover)

~ (Author) "The world of Euclidean geometry is inhabited by lines and planes..." (more)
Key Phrases: shortest arclength, fixed endpoint problem, ruling patch, Weierstrass E-function, Newton's Law, Curvatures Exercise (more...)

Editorial Reviews

Product Description

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.

Book Description

This book studies the differential geometry of surfaces and aims to help students make the transition from the standard university curriculum to a type of mathematics that is a unified whole, by mixing geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences.

See all Editorial Reviews

Product Details

• Hardcover: 510 pages
• Publisher: The Mathematical Association of America; 2 edition (July 10, 2007)
• Language: English
• ISBN-10: 0883857480
• ISBN-13: 978-0883857489
• Product Dimensions: 9.8 x 6.9 x 1.4 inches
• Shipping Weight: 2.6 pounds (View shipping rates and policies)
• Average Customer Review: 3.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon.com Sales Rank: #779,070 in Books (See Bestsellers in Books)

  Popular in this category: (What's this?)

  #100 in

  Books > Professional & Technical > Professional Science > Mathematics > Geometry & Topology > Differential Geometry

Customer Reviews

Most Helpful Customer Reviews

17 of 19 people found the following review helpful:

5.0 out of 5 stars clearest undergrad differential geometry text around, April 20, 2005

By	A. Ali "Harkonnen" (Minneapolis, MN USA) - See all my reviews (REAL NAME)

This review is from: Differential Geometry and Its Applications (2nd Edition) (Hardcover)

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.

6 of 6 people found the following review helpful:

4.0 out of 5 stars Nice introduction and applications of differential geometry, May 21, 2004

By A Customer

This review is from: Differential Geometry and Its Applications (2nd Edition) (Hardcover)

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.

10 of 23 people found the following review helpful:

2.0 out of 5 stars Not a text for a rigorous mathematics course, October 29, 1999

By A Customer

This review is from: Differential Geometry and Its Applications (Hardcover)

This book is not to be used as a rigorous introduction to differential geometry. There are some definitions and theorems that are casually described, and the motive behind particular definitions are vague. Those not interested in MAPLE might find constant instructions for MAPLE annoying. Not to be completely negative, there are some good excercizes in the text that I especially enjoyed.