N-Dimensional Spheres

27.10.23

This week I thought about calculating the limit of functions which go from $R^n$ to $R^m$ because it is very interesting to see how they behave for some $x \in R^n$ when $\|x\|$ goes to zero.

In $R^2$ there is that interesting trick where you use the trigonometic functions und then you only have one variable which you then can let move to zero. So that you only have to deal with one variable in you limit expression.

Definition

If you always want to have only one parameter in your limit expression you always want to approximate your limit with a sphere, where the center of it is your limit and the radius is the delta.

Spheres have a pretty simple property:

$\begin{equation*} S_r^n = \left\{ \|x\|_n = r \mid x \in R^n \right\} \end{equation*}$

Where $r$ is the radius and $n$ is the number of dimensions.

Trigonometric Representation

We now want to prove that the following set defines some arbitrary sphere $S_r^n$ . To do so we define $S_r^n$ recursivly:

$\begin{align*} S^1_r &= \left\{ r \right\} \\ S_r^n &= \left\{ r \cdot \begin{pmatrix} cos(\alpha_n) \cdot x_1 \\ cos(\alpha_n) \cdot x_2 \\ ...\\ cos(\alpha_n) \cdot x_{n-1} \\ sin(\alpha_n) \end{pmatrix} \mid \begin{pmatrix} x_1 \\ x_2 \\ ... \\ x_{n - 1} \end{pmatrix} = S_{n - 1, 1} \right\} \end{align*}$

Squaring Properties

So right now we only need to prove that the definition of our $S_r^n$ above holds the definition of a sphere. But first we will prove that

$\begin{equation*} \sum_{y \in U^n} y^2 = 1 \end{equation*}$

where $U^n = S^n_1$ to be more readable. We will prove now this hypothesis with induction.

Base Case

Our Base case is $n = 1$ so it follows that $\sum_{y \in U^1} = 1$ by the definition of $U^1$

Induction Step

$\begin{align*} & \sum_{y \in U^{n + 1}} y^2 \\ (\text{def. of $U^{n+1}$}) & = cos(\alpha_{n+1})^2 (\sum_{y \in U^n} y^2) + sin(\alpha_{n+1})^2 \\ (\sum_{y \in U^n} y^2 = 1) & \Rightarrow cos(\alpha_{n+1})^2 + sin(\alpha_{n+1})^2 = 1 \\ &&\square \end{align*}$

Prove of Trigonometric Representation

Given some $x \in S_r^n$ we need to show that $\|x\|_n = r$ which can be also written as $\|x\|_{n}^2 = r^2$ .

$\begin{align} \|x\|_{n}^2 &= \sqrt{r^2 (cos(\alpha_n)^2 x_1^2 + ... + cos(\alpha_n)^2 x_{n-1}^2 + sin(\alpha_n)^2)}^2 \\ &= r^2 (cos(\alpha_n)^2 x_1^2 + ... + cos(\alpha_n)^2 x_{n-1}^2 + sin(\alpha_n)^2) \\ &= r^2 ( cos(\alpha_n)^2 (x_1^2 + ... + x_{n - 1}^2) + sin(\alpha_n)^2) \\ &= r^2 ( cos(\alpha_n)^2 (\sum_{y \in U^{n - 1}} y^2) + sin(\alpha_n)^2) &&\text{ (Squaring Prop.)} \\ &= r^2 ( cos(\alpha_n)^2 + sin(\alpha_n)^2) &&\text{ ($cos(\alpha)^2 + sin(\alpha)^2 = 1$)} \\ &= r^2 \\ && \square \end{align}$

Use Case

Right now the only use-case I can find for this formula, is to find the limit of a N-Dimensional function. But the problem with this representation is that it gets quiet long because the number of $cos(\alpha_i)$ which are multiplied to $x_j$ grow proportional to $n$ in $S_r^n$ . Maybe there is some way to structure the expression to make it more useful and powerful for real use-cases.