Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts

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Notes

1 Assuming a perfectly spherical Earth.

2 As I write these words, I am looking at a copy of do Carmo’s Differential Geometry of Curves and Surfaces [4] with a broken spine—the result of my throwing it across the room almost two decades ago in sheer frustration at the section in which the Theorema Egregium is proved.

3 I will gesture here in the direction of Linderholm’s Mathematics Made Difficult [12], though with the warning that some of the fictional vignettes are regrettable.

4 Indeed, spherical geometry is now recognized as a classic non-Euclidean geometry, but seems not to have been seriously considered as such prior to the 19th century because the sphere’s geometry was apparently induced by the Euclidean 3-space in which it sat. It was only the Theorema Egregium and related results which showed that spherical geometry can be self-contained.

5 I never expected to see an explanation of neap tides in a differential geometry book, but now I have.

6 This review is also notable because it seems to be where domain coloring of complex functions was first introduced.