Elementary Differential Geometry

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them are accessible to higher level undergraduates. Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there. The book will prove an invaluable resource to all those taking a first course in differential geometry, for their lecturers, and for all others interested in the subject. Curves And Surfaces Are Objects That Everyone Can See, And Many Of The Questions That Can Be Asked About Them Are Natural And Easily Understood. Differential Geometry Is Concerned With The Precise Mathematical Formulation Of Some Of These Questions, And With Trying To Answer Them Using Calculus Techniques. It Is A Subject That Contains Some Of The Most Beautiful And Profound Results In Mathematics Yet Many Of These Are Accessible To Higher-level Undergraduates. Elementary Differential Geometry Presents The Main Results In The Differential Geometry Of Curves And Surfaces While Keeping The Prerequisites To An Absolute Minimum. Nothing More Than First Courses In Linear Algebra And Multivariate Calculus Are Required, And The Most Direct And Straightforward Approach Is Used At All Times. Numerous Diagrams Illustrate Both The Ideas In The Text And The Examples Of Curves And Surfaces Discussed There. The Book Will Provide An Invaluable Resource To All Those Taking A First Course In Differential Geometry, For Their Lecturers, And For All Others Interested In The Subject. Andrew Pressley Is Professor Of Mathematics At King’s College London, Uk. The Springer Undergraduate Mathematics Series (sums) Is A Series Designed For Undergraduates In Mathematics And The Sciences Worldwide. From Core Foundational Material To Final Year Topics, Sums Books Take A Fresh And Modern Approach And Are Ideal For Self-study Or For A One- Or Two-semester Course. Each Book Includes Numerous Examples, Problems And Fully Worked Solutions. 1. Curves In The Plane And In Space -- 2. How Much Does A Curve Curve? -- 3. Global Properties Of Curves -- 4. Surfaces In Three Dimensions -- 5. The First Fundamental Form -- 6. Curvature Of Surfaces -- 7. Gaussian Curvature And The Gauss Map -- 8. Geodesics -- 9. Minimal Surfaces -- 10. Gauss’s Theorema Egregium -- 11. The Gauss-bonnet Theorem -- Solutions -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11. By Andrew Pressley. Curves and surfaces are objects that you can see, and many of the questions that can be asked about them are natural and easily understood. This work presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. If asked to give an example of a curve, you might give a straight line, say y - 2x = 1 (even though this is not 'curved'!), or a circle, say x2 + y2 = 1, or perhaps a parabola, say y - x2 = 0.