\chapter{Differential forms}

The differential offers a notion of a derivative independent of local charts thanks to tangent spaces. Given that the bread and butter of mathematical analysis is differentiation and integration, we're naturally lead to the following question; can a notion of an integral be developed independent of local charts? 


To do this, we mix the concept of an exterior algebra with tangent spaces.

\begin{definition}[Differential $k$-form]
	$\bigwedge_{k} T^{*}_{x}M$
\end{definition}

We can think of differential $k$-forms as alternating covectors of the tangent space with $k$ parameters.


\begin{definition}[Smooth ifferential $k$-form]
	$\bigwedge_{k} T^{*}_{x}M$
\end{definition}


\begin{definition}[1-form]
	Function between 
\end{definition}


\section{Exterior derivative}
\[d \omega = \sum d\]