Exploring Curvature

1. Front Matter

   Pages i-xv

   PDF

2. The Evolution of Geometry

   • James Casey

   Pages 1-34

3. Basic Operations

   • James Casey

   Pages 35-40

4. Intersecting with a Closed Ball

   • James Casey

   Pages 41-44

5. Mappings

   • James Casey

   Pages 45-57

6. Preserving Closeness: Continuous Mappings

   • James Casey

   Pages 58-66

7. Keeping Track of Magnitude, Direction and Sense: Vectors

   • James Casey

   Pages 67-79

8. Curves

   • James Casey

   Pages 80-91

9. Arc Length

   • James Casey

   Pages 92-99

10. Tangent

    • James Casey

    Pages 100-112

11. Curvature of Curves

    • James Casey

    Pages 113-136

12. Surfaces

    • James Casey

    Pages 137-153

13. Surface Measurements

    • James Casey

    Pages 154-187

14. Intrinsic Geometry of a Surface

    • James Casey

    Pages 188-192

15. Gauss (1777-1855)

    • James Casey

    Pages 193-202

16. Normal Sections

    • James Casey

    Pages 203-222

17. Gaussian Curvature

    • James Casey

    Pages 223-232

18. Riemann (1826–1866)

    • James Casey

    Pages 233-244

19. Levi-Civita (1873–1941)

    • James Casey

    Pages 245-249

20. Parallel Transport of a Vector on a Surface

    • James Casey

    Pages 250-263

. . . one should not be too ready to erect a wall of separation between nature and the human mind. d'Alembert [Dugas (1955)] It is possible to present mathematics in a purely fonnal way, that is to say, without any reference to the physical world. Indeed, in the more advanced parts of abstract algebra and mathematical logic, one can pro­ ceed only in this manner. In other parts of mathematics, especially in Euclidean geometry, calculus, differential equations, and surface ge­ ometry, intimate connections exist between the mathematical ideas and physical things. In such cases, a deeper (and sometimes quicker) under­ standing can be gained by taking advantage of these connections. I am not, of course, suggesting that one should appeal to physical intuition whenever one gets stuck in a mathematical proof: in proofs, there is no substitute for rigor. Rather, the connections with physical reality should be made either to motivate mathematical assumptions, or to introduce questions out of which theorems arise, or to illustrate the results of an analysis. Such interconnections are especially important in the teaching of mathematics to science and engineering students. But, mathematics students too have much to gain by familiarizing themselves with the interconnections between ideas and real things. The present book explores the geometry of curves and surfaces in a physical way.

• Gaussian curvature
• commonplace curved objects
• curvature
• euklidische Geometrie
• experiments
• geometry
• moderne Geometrie

• Department of Mechanical Engineering, University of California, Berkeley, Berkeley, USA

  James Casey

Dr. J. Casey ist Professor an der University of California, Berkeley Department of Mechanical Engineering.

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