Titlebook: Manifolds, Vector Fields, and Differential Forms; An Introduction to D Gal Gross,Eckhard Meinrenken Textbook 2023 The Editor(s) (if applica

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Gal Gross,Eckhard Meinrenkeno Nuprl in order to prove a version of Brouwer’s continuity principle, as well as choice sequences in order to prove truncated versions of the axiom of choice and of Brouwer’s bar induction principle. This paper illustrate the process of extending Nuprl with versions of the axiom of choice.

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Manifolds,One of the goals of this book is to develop the theory of manifolds in intrinsic terms, although we may occasionally use immersions or embeddings into Euclidean space in order to illustrate concepts. In physics terminology, we will formulate the theory of manifolds in terms that are “manifestly coordinate-free.”

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Smooth Maps,A real-valued function on an open subset . is called . . ∈ . if it is infinitely differentiable on an open neighborhood of .. It is called . . if it is smooth at all points of .. The notion of smooth functions on open subsets of Euclidean spaces carries over to manifolds: A function is smooth if its expression in local coordinates is smooth.

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Submanifolds,Let . be a manifold of dimension .. We will define a .-dimensional submanifold . ⊆ . to be a subset that looks locally like ., regarded as the coordinate subspace defined by .. = ⋯ = .. = 0.

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Vector Fields,A vector field on a manifold may be regarded as a family of tangent vectors .. ∈ ... for . ∈ ., depending smoothly on the base points . ∈ .. One way of making precise what is meant by “depending smoothly” is the following.