What does $dS(x)$ mean in this integration? I mean, what is the definition of the following value?
$$I=\int_{\partial B(x,\, r)} u(y) dS(y)$$
Here, $u$ is a real-valued $C^2$ function on an open set $U$ in $\mathbb R^k$, and $B(x,\, r)$ is an open ball whose closure is contained in $U$. ($x$ is fixed.)
I have to replace the variable $y$ by $y=x+rz$ to get
$$I=\int_{\partial B(0,\,1)} u(x+rz) dS(z).$$
But this substitution confuses me. That $u(y)=u(x+rz)$ is alright. No problem with this. But why $dS(y)=dS(z)$? No need for the scalar $r^{n-1}$ multiplied? (The dimension of the $\partial B(x,\,r)$ is $n-1$.)
I'm not familiar with the notations like $dS(y)$, so it would be helpful if the answer provides an alternative description (e.g. the 'parametrized form' or anything) of the above integral.
 A: The notation $dS$ means the "induced surface measure" on the boundary of the ball. The notation $I=\int_{\partial B(x,r)}u(y)\,dS(y)$ is meant to mimic the measure-theory notation for integrals which is usually written as $\int_Xf(x)\,d\mu(x)$ (this is the notation used frequently in computations as opposed to the more concise $\int_Xf\,d\mu$ which is used for more theoretical purposes).
It is just notation to say "$y$ is the variable". Of course, this by itself has no meaning, because one is free to choose any letter they wish; all of the following mean the same thing:
\begin{align}
I=\int_{\partial B(x,r)}u\,dS = \int_{\partial B(x,r)}u(y)\,dS(y)=
\int_{\partial B(x,r)} u(\xi)\,dS(\xi)=
\int_{\partial B(x,r)} u(\ddot{\smile})\,dS(\ddot{\smile}).
\end{align}
And you're right there should be an $r^{n-1}$ when you make the change of variables $y=x+rz$, assuming of course, that the second $dS$ refers to the surface measure/area element on the unit sphere (this is of course the most meaningful interpretation of the notation... though if one wishes to be super precise, we could write something like $dS_r$ to mean the surface measure on the sphere of radius $r$ vs $dS_1$ for the sphere of unit radius... but I think this is overboard with the notation).

For example, in the case of $n=3$, we can parametrize the unit sphere using spherical coordinates $(\theta,\phi)\mapsto (\sin\theta \cos\phi,\sin\theta\sin\phi,\cos\theta)$, where $(\theta,\phi)\in(0,\frac{\pi}{2})\times (0,2\pi)$. In this case, $dS=\sin\theta\, d\theta\,d\phi$. Of course, this parametrization doesn't cover the entire sphere, but it covers "most of it" (i.e up to a set of measure zero), so for the purposes of integration this is perfectly suitable.
If $n=2$ we're just parametrizing the unit circle, in which case $dS$ is really just the $n-1=2-1=1$ dimensional "surface" element; i.e the line element, so this is just $d\phi$. Finally, if $n>3$, we can still use spherical coordinates to parametrize the sphere, but it gets more messy.
