I was trying to compute the area of the sphere using calculus and my knowledge of differential form as follow :
Consider the two form $\omega = dx \wedge dy$, we want to use this form to find out the area of a disk. We define a parametrization of the sphere as follows $F(r,\phi) \rightarrow (rcos\phi, rsin\phi)$ So we have :
\begin{align}
\int_{\mathbb{S}^2} \omega = \int_{[0,R]}\int_{[0,2\pi]} F^* \omega = \\
\int_{[0,R]}\int_{[0,2\pi]}rdr \wedge d\phi = \\
\int_{[0,R]}\int_{[0,2\pi]} rdrd\phi = \pi R^2
\end{align}
I was hoping to apply the same principle to find the circumference of the circle but I think I ultimately miss some technicalities.
The circumference of the circle is a 1-dimensional manifold so I am trying to define a one-form on it.I do not know how to proceed from here. Do I have to find a 1-d parametrization of the circle ? Is the one form $dx$ the one to integrate ?
I'm trying to compute the circumference of the circle using pullback an forms. So I can understand the mechanics of integrating forms to find volumes ( and hopefully move to more exotic manifolds like the area or volume of a torus ).
EDIT :
My reasoning is as follows. To find the circumference of a circle, let's define a mapping $F:\mathbb{R} \rightarrow \mathbb{R}$ that parametrize it. I was thinking of defining $F:[0,\frac{\pi}{2}] \rightarrow \mathbb{R}$ and multiply the result by 4 ( since the mapping parametrize a quarter of the circle ). Then proceed to integrate $\int_{[0,\frac{\pi}{2}]} F^*(dx)$ Unfortunately the few mapping I tried fail to provide me the right answer.
EDIT2 : Using stereo coordinates (attempt)
Let's consider the map $F:[0,\frac{\pi}{4}] \rightarrow \mathbb{R}$ the stereo projection of $\frac{1}{8}$ of the circumference of the circle, defined as $F(\alpha)=Rtan(\alpha)$ Now we have $\int_{\mathbb{\frac{S^1}{8}}} dx = \int_{[0,\frac{\pi}{4}]} F^*(d\alpha) = R\frac{\pi}{4}$ which when multiplied by 8 gives us the result $2\pi R$
Best Answer
$\newcommand{\Reals}{\mathbf{R}}$Stereographic coordinates give a more or less uniform way to approach the calculation of the $n$-volume of the unit $n$-sphere. If $x = (x_{1}, \dots, x_{n})$ denotes the general element of $\Reals^{n}$, the stereographic parametrization of the $n$-sphere is $$ f(x) = \frac{(2x, \|x\|^{2} - 1)}{\|x\|^{2} + 1} = \left(\frac{2x_{1}}{1 + \sum_{j} x_{j}^{2}}, \dots, \frac{2x_{n}}{1 + \sum_{j} x_{j}^{2}}, \frac{-1 + \sum_{j} x_{j}^{2}}{1 + \sum_{j} x_{j}^{2}}\right). $$ A straightforward (and edifying) calculation shows that the induced metric on $\Reals^{n}$ is $$ g = \frac{4(dx_{1}^{2} + \cdots + dx_{n}^{2})}{1 + \sum_{j} x_{j}^{2}}. $$ The volume form is therefore $$ dV = \left(\frac{2}{1 + \sum_{j} x_{j}^{2}}\right)^{n} dx_{1} \wedge \cdots \wedge dx_{n}. $$ The volume of the unit $n$-sphere $S^{n} \subset \Reals^{n+1}$ is the integral of this volume form over $\Reals^{n}$.
For instance,
The volume (i.e., length) of the unit circle is $$ \int_{-\infty}^{\infty} \frac{2\, dx}{1 + x^{2}}. $$ The value of this integral is
The volume (i.e., area) of the unit $2$-sphere is $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{4\, dx\, dy}{(1 + x^{2} + y^{2})^{2}}, $$ whose value (by converting to polar coordinates) is
To calculate these for arbitrary $n$, it may be easier to calculate the volume of the ball of radius $r$ (using Cavalieri's theorem and recursion, see for example Where does this formula for the volume of a $n$-dimensional ball come from? or Wikipedia's page on the $n$-sphere), then use the fact that if the $n$-ball has volume $V_{n}(r) = V_{n}(1)r^{n}$, then the $(n-1)$-sphere has $(n-1)$-dimensional volume $$ \frac{d}{dr} V_{n}(r) = nV_{n}(1)r^{n-1}. $$
You may also enjoy attempting the calculation in generalized spherical coordinates, a nice application of the reduction formula for integrating powers of sine.