Curvature in Mathematics and Physics (Dover Books on

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There's a lot of formalae and transformations which tell you how justified such things are and you can see just from thinking about it geometrically that while the approximation that the surface of the Earth is a cylinder is valid very close to the equator (ie your phi' ~ phi/sin(theta) ~ phi, since theta = pi/2), becomes more and more invalid as you go towards the poles. Also central to geometry this century has been the relation between analysis on manifolds (for example properties of the Laplace operators) and their topology and geometry.

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