Differential Forms

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,…,xⁿ, 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.