Surface integration w.r.t Hausdorff measure

Question

I am reading at Evans' book Measure Theory and Fine Properties of Functions, Revised Edition, p. 165 and I can't see how one gets the transition from the first dotted (🔴) integral to the second one --

We have

-- here $Df$ is just the gradient of $f$ and $d H^{n - 1}$ is the Hausdorff surface measure.

Any help would be much appreciated!

Answer 1

$w\in \partial B(z,ts)$, so $|w-z|=ts$ and hence $$ \frac{s^p}{t^{n-1}}=s^{n+p-1}|w-z|^{1-n}. $$ You can also check Lemma 2.33 in my Lecture notes, where the proof of the result you are studying is presented in a slightly different way.