After establishing some notational conventions which will be used throughout the book, we will begin with the notion of a differentiable manifold. These are spaces which are locally like Euclidean space and which have enough structure so that the basic concepts of calculus can be carried over. In this first chapter we shall primarily be concerned with the analogs and implications for manifolds of the fundamental theorems of differential calculus. Later, in Chapter 4, we shall consider the theory of integration on manifolds.
KeywordsVector Field Tangent Vector Open Cover Coordinate Function Integral Manifold
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