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The Geometry of Physics by Theodore Frankel

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CHAPTER 1

Manifolds and Vector Fields

image

Better is the end of a thing than the beginning thereof.

Ecclesiastes 7:8

 

As students we learn differential and integral calculus in the context of euclidean space imagen, but it is necessary to apply calculus to problems involving “curved” spaces. Geodesy and cartography, for example, are devoted to the study of the most familiar curved surface of all, the surface of planet Earth. In discussing maps of the Earth, latitude and longitude serve as “coordinates,” allowing us to use calculus by considering functions ...

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