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*

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s