Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X\_1, \dots, X\_k)$, the first $k < n$ coordinates of $X$ has density $p(x\_1, \dots, x\_k)$ with respect to Lebesgue measure in $\mathbf{R}^k$, moreover if $r^2 = x\_1^2 + \cdots + x\_k^2$, then it is proportional to 
$$
\left(1 - \frac{r^2}{n}\right)^{(n-k)/2 - 1}, \quad \text{if}~0 \leq r^2 \leq n,
$$
and otherwise is 0. I tried to compute this using the fact that $(X\_1, \dots, X\_k) \stackrel{\rm d}{=} \sqrt{n} (g\_1, \dots, g\_k)/\sqrt{g\_1^2 + \cdots + g\_n^2}$, when $g\_i$ are iid standard normal variables, but it was somewhat unclear to me how to use even this representation to compute the density. Can anyone sketch the details for me?



Marginal density of uniform spherical distribution
$\newcommand{\R}{\mathbb{R}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\X}{\mathbf{X}}$
This is to present a formalization of the answer by Carlo Beenakker, without explicit use of the delta function. 


We are going to assume that $\X=(X\_1,\dots,X\_n)$ is uniformly distributed on the unit sphere $\mathbb S^{n-1}$, rather than on $\sqrt n\,\mathbb S^{n-1}$.


For each real $t\in(0,1)$, define the measures $\mu\_t$ and $\nu\_t$ over $\R^n$ by the conditions 
\begin{equation\*}
 \int f\,d\mu\_t=\int\_{\R^n}d\x f(\x)1\_{1-t<|\x|^2\le1}
\end{equation\*}
and 
\begin{equation\*}
 \int f\,d\nu\_t=\frac{\int f\,d\mu\_t}{\int d\mu\_t}
\end{equation\*}
for all (say) nonnegative continuous functions $f\colon\R^n\to\R$, where $|\cdot|$ denotes the Euclidean norm. Then $\nu\_t$ is a probability measure converging (as $t\downarrow0$) to the Haar measure $h$ on the unit sphere in $\R^n$, in the sense that (say) 
\begin{equation\*}
 \int f\,d\nu\_t\to\int f\,dh 
\end{equation\*}
for all nonnegative continuous functions $f\colon\R^n\to\R$. 


Take now any function $f\colon\R^n\to\R$ such that
\begin{equation\*}
 f(\x)=g(\x\_{n-1})
\end{equation\*}
for some nonnegative continuous function $g\colon\R^{n-1}\to\R$ and all $\x\in\R^n$, where $\x\_j:=(x\_1,\dots,x\_j)$ for $\x=(x\_1,\dots,x\_n)\in\R^n$ and $j=1,\dots,n-1$. 
Then  

\begin{align\*}
 \int f\,d\mu\_t&=\int\_{\R^{n-1}}d\x\_{n-1}\,g(\x\_{n-1})\int\_\R du\, 1\_{1-t<|\x\_{n-1}|^2+u^2\le1} \\ 
 &=\int\_{\R^{n-1}}d\x\_{n-1}\,g(\x\_{n-1})
 (1+o(1))t\,(1-|\x\_{n-1}|^2)^{-1/2}\,1\_{|\x\_{n-1}|<1}. 
\end{align\*}
Also, 
\begin{equation\*}
 \int d\mu\_t=\int\_{\R^n}d\x\, 1\_{1-t<|\x|^2\le1}\propto(1+o(1))t, 
\end{equation\*}
where $\propto$ means an equality up to a constant factor, depending only on $n,k$. 
So,
\begin{equation\*}
 \int f\,dh=\lim\_{t\downarrow0}
 \int f\,d\nu\_t\propto\int\_{\R^{n-1}}d\x\_{n-1}\,g(\x\_{n-1})
 (1-|\x\_{n-1}|^2)^{-1/2}\,1\_{|\x\_{n-1}|<1}. 
\end{equation\*}
Thus, the joint pdf of $\X\_{n-1}=(X\_1,\dots,X\_{n-1})$ is given by 
\begin{equation\*}
 p\_{n-1}(\x\_{n-1})\propto(1-|\x\_{n-1}|^2)^{-1/2}\,1\_{|\x\_{n-1}|<1}.
\end{equation\*}
Now successively integrating $p\_{n-1}(\x\_{n-1})$ ($n-1-k$ times) in $x\_{n-1},\dots,x\_{k+1}$ and each time using the formula 
\begin{equation\*}
 \int\_0^{b^{1/2}}(b-u^2)^p du=c\_p b^{p+1/2}
\end{equation\*}
for real $b>0$ and $p>-1$ with $c\_p:=\int\_0^1(1-u^2)^pdu\in(0,\infty)$ (so that $1/2$ is added to the exponent $p$ after such an integration), we see that the joint pdf of $\X\_k=(X\_1,\dots,X\_k)$ is given by 
\begin{equation\*}
 p\_k(\x\_k)\propto(1-|\x\_k|^2)^{(n-k)/2-1}\,1\_{|\x\_k|<1},
\end{equation\*} 
as desired. 


