# Differential Forms

1. Dec 30, 2009

### Nusc

1. The problem statement, all variables and given/known data
Let $$x_1,...,x_n: M \rightarrow R$$ be functions on a manifold which form a local coordinate system on some region. Show that every differential form on this region can be written uniquely in the form

$$w^k = \sum_{i_1<...<i_k} a_{i_1,...i_k}(\bf{x})dx_{1_i} \wedge .. \wedge dx_{1_k}$$

Any ideas?

2. Relevant equations

3. The attempt at a solution

2. Dec 30, 2009

### HallsofIvy

Staff Emeritus
Suppose you are in 3 dimensions. How would you write $d_{x_1}d_{x_3}$ in that form?