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Ryuma ORITA
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Geometry IB

Term 2, Mon. Period 3 & Thu. Period 2 · Faculty of Science, Niigata University

Note

This course is held online via Zoom.

References

  1. Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications
  2. M. Kurita, “復刊 積分幾何学” (in Japanese), Kyoritsu Shuppan
  3. H. Kawasaki, “極値問題” (in Japanese), Yokohama Tosho

Course contents

  1. History of the isoperimetric problem; an elementary-geometric “proof” (06-18)
  2. The isoperimetric inequality (a proof via calculus) (06-22)
  3. Coordinates on the set of lines in the plane; the measure of sets of lines (06-25)
  4. The Cauchy–Crofton formula (06-29)
  5. Applications of the Cauchy–Crofton formula (07-02)
  6. Coordinates on the set of positions in the plane; the measure of sets of positions (07-06)
  7. The isoperimetric inequality (a proof using sets of positions) (07-09)
  8. The isoperimetric inequality (Blaschke’s proof) (07-13)
  9. Poincaré’s formula for sets of positions (07-16)
  10. The area of the region bounded by parallel curves (07-20)
  11. The volume of tubes (07-23)
  12. The isoperimetric inequality (Santaló’s proof) (07-27)
  13. The fundamental problem of the calculus of variations (07-30)
  14. The brachistochrone problem (08-03)
  15. Constrained variational problems (08-06)
  16. The isoperimetric inequality (a proof via the calculus of variations); the inscribed rectangle problem (08-10)