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I am reading an article in which real functions on the 2-dimensional sphere (in $\mathbb{R}^3$), $f:\mathbb{S}^2 \to \mathbb{R}$, are considered and analyzed. The analysis discusses the spherical gradient $\nabla f$, the Laplacian $\Delta f$, and the Hessian matrix $\nabla \nabla f$, as well as a form of Taylor's theorem using the spherical gradient and Hessian in place of first and second derivatives.

I've understood these are defined by considering an atlas of two charts on the sphere, translating to local coordinates, and then taking the gradient, Laplacian and Hessian in the usual way ("on $\mathbb{R}^2$").

I'm looking for a book that explains how this analysis is done. I am currently only interested in two manifolds: the sphere (maybe in higher dimensions) and the flat torus, so a general theory of analysis on Riemann manifolds might be too much for right now. This subject does not seem to appear in Baby/Big Rudin, but any book similar (in level of difficulty) to those would be perfect.

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You could consider any book on smooth manifolds. I strongly endorse the following two books:

  1. An introduction to Manifolds by L.W. Tu.
  2. An Introduction to Smooth Manifolds by J.M. Lee.
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I don't know of any book dedicated to just $S^2$ and $T^2$, but I think that spivak's book "Analysis on manifolds" is a good and readable introduction.

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