# de Rahm Complexes: Really cool math, or a miracle?

Before explaining the title, here’s a little background:

Let $\Omega^*$ be the vector space over $\mathbb{R}$ with basis

$1,dx_i,dx_idx_j,dx_idx_jdx_k,\ldots,dx_l\ldots dx_n$.

Using this notation, the collection $\Omega^*(\mathbb{R}^n)$ of $C^\infty$ differential forms on $\mathbb{R}^n$ are elements

$\begin{array}{rcl}\Omega^*(\mathbb{R}^n) & = & \left\{C^\infty\text{ functions on }\mathbb{R}^n\right\}\otimes_{\mathbb{R}}\Omega^*\\[0.5em] & = & \oplus_{q=0}^n \Omega^q(\mathbb{R}^n)\end{array}$

where $\Omega^q(\mathbb{R}^n)$ consists of $C^\infty$ $q$-forms on $\mathbb{R}^n$. There also exists a differential operator $d:\Omega^q(\mathbb{R}^n)\to\Omega^{q+1}(\mathbb{R}^n)$ which satisfies the expected properties for exterior differentation. Under this construction, the pair $\left(\Omega^*(\mathbb{R}^n),d\right)$ is called the de Rahm complex on $\mathbb{R}^n$. Moreover, the kernel and image of $d$ are known as the closed and exact forms, respectively.

When this material was presented in Bott & Tu’s Differential Forms in Algebraic Topology, the following quote was included:

The de Rahm complex may be viewed as a God-given set of differential equations, whose solutions are the closed forms. For instance, finding a closed 1-form $f\,dx+g\,dy$ on $\mathbb{R}^2$ is tantamount to solving the differential equation $\partial g/\partial x-\partial f/\partial y=0$….

So maybe there is a God, and maybe God is a mathematician? *ponders*