This is a doubt that I carry since my PDE classes.
Some background (skippable):
In the multivariable calculus course at my university we made all sorts of standard calculations involving surface and volume integrals in $R^3$, jacobians and the generalizations of the fundamental theorem of calculus. In order to make those calculations we had to parametrize domains and calculate differentials.
A couple of years later I took a PDE course. We worked with Evans' Partial differential equations book. This was my first experience with calculus in $\mathbb R^n$ and manipulations like
$$\text{average}\int_{B(x,r)}f(y)\,dy= \text{average}\int_{B(0,1)}f(x+rz)\,dz.$$
This was an ordinary change of variables. $y=x+rz,\,\,dy=r^n\,dz$ and the mystery was solved. Like in that case, I was able to justify most of these formal manipulations after disentangling definitions.
That aside, I found these quick formal calculations to be very powerful.
However,
I realized that I wasn't able to justify this:
$$\text{average} \int_{\partial B(x,r)}f(y)dS(y)= \text{average}\int_{\partial B(0,1)}f(x+rz)\,dS(z).$$
I have some vague idea of what's happening: the same substitution as before, but this time the jacobian is $r^{n-1}$ because the transformation is actually happening between regions which "lack one dimension". Also, I see some kind of pattern: a piece of arc-length in the plane is $r\,d\theta$, a piece of sphere-area is $r^2 \sin\theta \, d\phi \,d\theta$, "and so on". Maybe some measure-theoretic argument can help me: I know, roughly speaking, that for any measure $\mu$, $$\int_\Omega f\circ \phi \,d\mu=\int_{\phi(\Omega)} f \, d(\mu\circ\phi^{-1}).$$ I'd say $\phi(z)=(z-x)/r$ and $\phi^{-1}(y)=ry+x$, but I actually don't know how $dS(y)$ looks like "as a measure" (It's not a product measure or a restriction of one, but it somehow relates to Lebesgue's in $\mathbb R^n$…). Why would I conclude that $dS(y)\circ \phi^{-1}=r^{n-1}dS(z)$? I have an intuition, but either I lack the mathematical concepts and definitions to express it or I'm just too confused. Is there some theory that I could learn in order to understand? Maybe something about the measure $dS$. Is it expressible in terms of the Lebesgue measure in some way? Or set-theoretically, maybe, without having to resort to $n-1$ parameters and complicated relations?
Maybe all of this would not have been a problem if I had ever mastered n-dimensional spherical coordinates. But even so, more generally, is there a way of changing variables when I'm integrating over a subregion of "dimension$<n$" without necessarily parametrizing?
Sorry for the vagueness, but I don't really know what to ask for exactly.
Note: I saw some of the answers to this post, but none of them were deep enough in the direction I intend.
Note II: If there are no general methods or theories, maybe restricting to linear transformations, to Lebesgue measure exclusively, or to subregions defined by simple expressions like $g(x)=C$ or $g(|x|)=C$ could get me somewhere.
Edit: I have not yet studied differential geometry, which has been mentioned in a comment. I added it to the tags.
Best Answer
I know this is an old question, but I thought this explanation might be helpful to some.
By definition (in $\mathbb R^3$):
$$\int_{\partial B(\boldsymbol x,r)}f(\boldsymbol y)dS(\boldsymbol y)= \int_U f(\boldsymbol y(s,t))\left\|\frac{\partial\boldsymbol y}{\partial s}\times\frac{\partial\boldsymbol y}{\partial t}\right\|dsdt$$
Now, observe that $f(\boldsymbol y)=f(\boldsymbol x+r(\frac{\boldsymbol y-\boldsymbol x}{r}))$, and that if $\boldsymbol y(s,t)$ is a parametrization of $\partial B(\boldsymbol x,r)$ for $(s,t)\in U$, then $\frac{\boldsymbol y(s,t)-\boldsymbol x}{r}$ is a parametrization of $\partial B(\boldsymbol 0,1)$ for $(s,t)\in U$. Finally we observe that
$$\left\|\frac{\partial\boldsymbol y}{\partial s}\times\frac{\partial\boldsymbol y}{\partial t}\right\|= r^2\left\|\frac{\partial}{\partial s} \left (\frac{\boldsymbol y-\boldsymbol x}{r} \right )\times\frac{\partial }{\partial t} \left (\frac{\boldsymbol y-\boldsymbol x}{r} \right )\right\|$$
So if we let $\boldsymbol z(s,t)=\frac{\boldsymbol y(s,t)-\boldsymbol x}{r}$, then we have
$$\int_U f(\boldsymbol y(s,t))\left\|\frac{\partial\boldsymbol y}{\partial s}\times\frac{\partial\boldsymbol y}{\partial t}\right\|dsdt= r^2\int_U f(\boldsymbol x +r\boldsymbol z(s,t))\left\|\frac{\partial\boldsymbol z}{\partial s}\times\frac{\partial\boldsymbol z}{\partial t}\right\|dsdt\\= r^2\int_{\partial B(\boldsymbol 0,1)}f(\boldsymbol x+r\boldsymbol z)dS(\boldsymbol z)$$
Edit by OP
As @user5753974 commented, you can generalize this if you use the fact that in $\mathbb R^n$ $$∫_{∂B(\boldsymbol x,r)}f(\boldsymbol y)dS(\boldsymbol y)=∫_{U}f(\boldsymbol y(\boldsymbol z)) \left \|\det\left (\frac{∂\boldsymbol y}{∂z_1},…,\frac{∂\boldsymbol y}{∂z_{n−1}},\boldsymbol n\right) \right \| d^{n−1}\boldsymbol z,$$ where $\boldsymbol n$ is the normal vector to the surface, and that $\boldsymbol n$ does not change when the surface is scaled and translated.