Linked Questions

I am acquainted with polar coordinates and their reason of being, that is: Taking a vector in $2-$dimensions, we can normalize it: $$\cfrac{1}{|(x,y)|}(x,y)=(x',y')$$ And we can clearly see that: $$...

Is there an extension to $n$ dimensions of the usual spherical coordinates mapping a three-dimensional sphere to a two-dimensional rectangle? [Duplicate]: Analogue of spherical coordinates in $n$-...

Possible Duplicate: Analogue of spherical coordinates in $n$-dimensions If we take a 2-sphere of radius a, we can define $ f(z, t) = (\sqrt{a^2-z^2}\cos t,\sqrt{a^2-z^2}\sin t, z) $ it's a ...

I thought about the parameterizing the 4D sphere using $$x_1= r\cos(\theta_1), x_2= r\sin(\theta_1)\cos(\theta_2), x_3= r\sin(\theta_1)\sin(\theta_2)\cos(\theta_3),\\ x_4= r\sin(\theta_1)\sin(\theta_2)...

I need to find the volume of the region defined by $$\begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...

I have found an answer on this site to the question of determining the mean straight-line distance between 2 randomly chosen points in a disc of radius r. (See Average distance between two points in a ...

What is the formal definition of the direction of a 3D vector? In the plane, it can be defined as the (obtuse) angle formed by the position vector and $(1,0)$. In 3D space, it is the angles formed ...

An $n\times n$ orthogonal real matrix $A$ is a set ${A_{ij}}$ of $n^2$ real numbers that satisfy the constraints: $$\sum_k A_{ik} A_{kj} = \delta_{ij} $$ for all $1\leq i,j\leq n$. The equations (1.) ...

Let $\alpha >0$ a real number and $k>0$ an integer. I wolud like to know for which $\alpha$ the multiple series $$\sum_{n_{1}=1}^{\infty}\cdots\sum_{n_{k}=1}^{\infty}\frac{1}{\left(n_{1}^{2}+\...

I have the radius of the multi-dimensional sphere plus angles for each dimension-1. I know I can use sin/cos functions to obtain coordinates, but is there a computationally faster way exists?