Needham, Chapter 7 Exercises for Act II, — PDF files with figures and answers for exercises

The figures and exercises in this list pertain to Visual Differential and Forms (2021) by Tristan Needham. The answers are my own and they are not endorsed by the author of VDGAF. Current discussions of the book and the exercises appear on Vasco's forum. The answers linked there are in general more consistent and reliable. I consult them frequently in my own studies.

  1. Coordinate independence of K (PDF)
  2. Central Projection, [4.1]. (PDF)
  3. Central Projection. Rederive (4.2) (PDF)
  4. Central Projection. Curvature from (4.10) and metric (PDF)
  5. Angular excess of n-gon (PDF)
  6. Stereographic Metric (PDF)
  7. Antipodal points (PDF)
  8. Conformal ("Isothermal") Coordinates (PDF)
  9. Conformal Curvature Formula (PDF)
  10. Archimedes-Lambert Projection (PDF)
  11. Central Cylindrical Projection (PDF)
  12. Mercator Projection (PDF)
  13. An Impossibly Good Map (PDF)
  14. Polya's Mechanical Proof of Snell's Law (PDF)
  15. Snell's Law and Velocity (PDF)
  16. Tractrix Parameterization (PDF)
  17. Cylindrical-Polar Area on Sphere (PDF)
  18. The pseudosphere has finite area! (PDF)
  19. Pseudosphere construction project, not included.
  20. Geodesics of the Pseudosphere. (PDF)
  21. Conformal Curvature Formula. (PDF)
  22. Surfaces of Revolution of Constant Curvature. (PDF)
  23. Curvature of the Torus. (PDF)
  24. Amplitwist of z^m. (PDF)
  25. Metric of the Beltrami-Poincare Disc. (PDF)
  26. Curvature of the Beltrami-Poincare Disc. (PDF)
  27. Moebius Rotations of the Riemann Sphere.
    • Parts (i-iii) (PDF)
    • Parts (iv-vii) (PDF)
    • Parts (viii-ix) (PDF)
  28. Metric of H^2 within H^3. (PDF)
  29. Curvature of Hemispherical H^2 in H^3. (PDF)
  30. Horosphere Metric and Curvature. (PDF)
  31. Spheres in H^3. (PDF)
  32. Horosphere Metric. (PDF)
  33. Penrose's Direct Labeling of Light Rays with Complex Numbers. (PDF)
  34. Einstein's Aberration Formula [In progress]. (PDF)
Last modified on October 18, 2022.

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