Differential Forms

  • Georges de Rham
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 266)

Abstract

On a manifold, we obtain differential forms of degree 1 as sums of the products of a function g by the differential df of another function f,
$$\sum {gdf.}$$
Expressing this in terms of the local coordinates x1,,x n , the above differential form reduces to the expression
$$\sum\limits_{i = 1}^n {{a_i}} d{x^i}\;with\;{a_i} = \sum {g\frac{{\partial f}}{{\partial {x^i}}}} .$$
If we change the local coordinate system, the coefficients a i transform as the components of a covector.

Keywords

Manifold 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Georges de Rham
    • 1
  1. 1.LausanneSwitzerland

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