# The Unapologetic Mathematician

## An Example (part 3)

Now we can take our differential form and our singular cube and put them together. That is, we can integrate the $1$-form $\omega$ over the circle $c_a$.

First we write down the definition:

$\int\limits_{c_a}\omega=\int\limits_{[0,1]}{c_a}^*\omega$

to find this pullback of $\omega$ we must work out how to push forward vectors from $[0,1]$. That is, we must work out the derivative of $c_a$.

This actually isn’t that hard; there’s only the one basis vector $\frac{d}{dt}$ to consider, and we find

$\displaystyle {c_a}_{*t}\left(\frac{d}{dt}\right)=-2\pi a\sin(2\pi t)\frac{\partial}{\partial x}+2\pi a\cos(2\pi t)\frac{\partial}{\partial y}$

We also have to calculate the composition

$\displaystyle\begin{aligned}\omega(c_a(t))&=-\frac{a\sin(2\pi t)}{(a\cos(2\pi t))^2+(a\sin(2\pi t))^2}dx+\frac{a\cos(2\pi t)}{(a\cos(2\pi t))^2+(a\sin(2\pi t))^2}dy\\&=\frac{1}{a}\left(-\sin(2\pi t)dx+\cos(2\pi t)dy\right)\end{aligned}$

This lets us calculate

$\displaystyle\begin{aligned}\int\limits_{c_a}\omega&=\int\limits_{[0,1]}{c_a}^*\omega\\&=\int\limits_{[0,1]}\left[\omega(c_a(t))\right]\left({c_a}_{*t}\left(\frac{d}{dt}\right)\right)\,dt\\&=\int\limits_{[0,1]}\left[\frac{1}{a}\left(-\sin(2\pi t)dx+\cos(2\pi t)dy\right)\right]\left(2\pi a\left(-\sin(2\pi t)\frac{\partial}{\partial x}+\cos(2\pi t)\frac{\partial}{\partial y}\right)\right)\,dt\\&=2\pi\int\limits_0^1\sin(2\pi t)^2+\cos(2\pi t)^2\,dt=2\pi\int\limits_0^1\,dt=2\pi\end{aligned}$

So, what conclusions can we draw from this? Well, Stokes’ theorem now tells us that the $1$-form $\omega$ cannot be the differential of any $0$-form — any function — on $\mathbb{R}^2\setminus{0}$. Why? Well, if we had $\omega=df$, then we would find

$\displaystyle\int\limits_{c_a}\omega=\int\limits_{c_a}df=\int\limits_{\partial c_a}f=\int\limits_{0}f=0$

which we now know not to be the case. Similarly, $c_a$ cannot be the boundary of any $2$-chain, for if $c_a=\partial c$ then

$\displaystyle\int\limits_{c_a}\omega=\int\limits_{\partial c}\omega=\int\limits_{\partial c}d\omega=\int\limits_{\partial c}0=0$

It turns out that there’s a deep connection between the two halves of this example. Further, in a sense every failure of a closed $k$-form to be the differential of a $k-1$-form and every failure of a closed $k$-chain to be the boundary of a $k+1$-chain comes in a pair like this one.

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Posted by John Armstrong | Differential Topology, Topology

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