## Introduction

Currently while visiting the “KoMa” (Konferenz der deutschsprachigen Mathematikfachschaften / conference of German-speaking mathematics students) I got into a lot of interesting conversations about geometry, topology and category theory. This set the stage for me to think about differential forms because I wanted to understand how exactly these things look and feel like.

## My sketches

```
Notes about differential forms
```

The proof mentioned on page 1 is viewable on https://web.archive.org/web/20211224033113/https://planetmath.org/tensorproductofdualspacesisadualspaceoftensorproduct

## Notation

Throughout this article we will choose the following symbols and notation:

- \(M\) a smooth manifold over a field \(K\subset \mathbb{R}\)
- \(p\in M\) a point in \(M\)
- \(T_p M\) the tangent space of \(M\) at \(p\)
- \(T_p^\ast M\) the cotangent space of \(M\) at \(p\), which consists of the linear functions \(\phi: T_p M \to K\). For finite tangent spaces \(T_p M\), we can canonically identify \(T_p^\ast M\) with \(T_p M\).
- \(T M = \bigcup_{p\in M} \{p\}\times T_p M\) and \(T^\ast M = \bigcup_{p\in M} \{p\}\times T_p^\ast M\) are the tangent/cotangent bundles of \(M\)
- \(\Lambda^k(T^\ast M)\) is the k-th exterior product of the cotangent bundle of \(M\). Its elements \(\mu_p\in\Lambda^k(T^\ast M)\) are called k-coblades and allow to measure volumes on some tangent space \(T_pM\) in \(k\)-dimensions.
- \(\Gamma(M, \Lambda^k(T^\ast M)) = \Gamma(\Lambda^k(T^\ast M))\) is the space of global sections from the basespace \(M\) to the total space \(\Lambda^k(T^\ast M)\). We use this, because for an element \(\omega:M \to \Lambda^k(T^\ast M)\) in it, we can smoothly associate for each point \(p\) on \(M\) a k-coblade.

## Formal definition

A differential form \(\omega\) is defined as an element in the space of global sections over the kth exterior algebra of the cotangent bundle over a smooth manifold \(M\). We denote this as:

\[\omega \in \Gamma(M, \Lambda^k(T^\ast M))\]

## My interpretation

Basically, a differential 1-form can be visualized as appending a vector onto each point of our manifold. Because we work with a smooth section, these vectors need to vary “smoothly” from one point to another. This means, if we wiggle the point we look at on our manifold just a tiny bit, the vector we have related to this point will also wiggle just a tiny bit. Also, our corresponding vector cannot take any sharp turns if we do not move our point in sharp turns.