# Differentiable Manifolds

• E. Hlawka
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 7/1970)

## Abstract

Let M be a set. An n-dimensional chart c on M is a pair (U,ϕ) , where U is a subset of M and ϕ a bijective mapping of U onto an open set of Rn . U is called the domain of c,ϕ the coordinate mapping of c. For each pεU we can write (see Fig. 1)
$${\rm{f(p)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{f}}_{{\rm{1}}\,}}{\rm{(p)}}\,{\rm{,}}...{\rm{,}}{{\rm{f}}_{\rm{n}}}\,{\rm{(p)}}\,{\rm{)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{x}}_{\rm{1}}}\,{\rm{,}}...{\rm{,}}{{\rm{x}}_{\rm{n}}})$$
We call (x1,...,xn) the local coordinates of p in the chart (U,ϕ). The notation can be made more explicit by writing x1,...,xn in the form
$$({x_1}\,(p)\,,...,{x_{n\,}}\,(p)\,)$$
or, in short, x(p)

## Keywords

Vector Field Coordinate Function Tensor Field Leibniz Rule Differentiable Manifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Literature

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