Let us have an $A \subset \mathbb{R^d}$, that is a $k$-dimensional $\mathcal{C}_1$ surface (obviously $k<d$) and let $a \in A$. Why then is $\Theta^k(A,a)=1?$
Of course $\Theta^k := \lim_{r \rightarrow 0} \frac{\mathcal{H}^k(A \cap B(a,r))^k}{r^k \omega_k},$ where $B(a,r)$ is an open ball centered in $a$ with a diameter $r$ and $\omega_k$ is a volume of a $k$-dimensional ball with a diameter $1$.
I know that one should probably find a good parametrization of $A$ that will have an orthonormal jacobi matrix in the point whose image is $a$, so its jacobian is $1$ and I can easily evaluate the surface integral over the small ball around $a$ (intersected with $A$) which is going to be exactly the volume of $k$-dimensional ball of the same diameter, but I am kind of clueless on how to do that. Could anyone perhaps explain to me how that parametrization can be found (assuming that this is the right approach to find the answer)?
I'm sorry for asking such a basic question, but since I mostly do probability theory and very little geometry (so far), I'm a little dumb in these things.
Thanks for the answers!