- Published 2010

Let r be coordinates on an open set in R and x be r ◦ φ. Definition 1.5 (C∞ function). f : M → R is C∞ at p if there exists (U, φ) with p ∈ U such that f ◦ φ−1 is C∞ at φ(p) ∈ R. Definition 1.6 (Partial Derivatives). If f ∈ C∞(M) and (U, x, . . . , x) is a chart containing p, then we define ∂ ∂xi |pf = ∂ ∂ri |φ(p)f ◦ φ −1 Definition 1.7 (Tangent Space). TpM , the tangent space at p, is the vector space with basis ∂ ∂xi at p ∈ (U, φ). Theorem 1.8. Let x and y be different sets of coordinates at a point p. Then

@inproceedings{2010TopologyC,
title={Topology , cohomology and sheaf theory},
author={},
year={2010}
}