Doubts on some notes on the co-area formula

Question

I am trying to read these notes on the co-area formula ( I wanted to understand better this A "contorted" Fubini. ) :

https://www3.nd.edu/~lnicolae/Coarea.pdf

but already at page 3 I have some doubts. Lemma 1 reads:

My doubts:

• First of all I guess that "Euclidean spaces" here are intended as vector spaces with a positive definite scalar product and "Euclidean coordinates" are intended as coordinates w.r.t. an orthogonal bases. Is this correct ?

• Lemma 1.1. isn't it just the standard Singular Value Decomposition ? Is there any difference ?

Than in claims Eq. 1.4 :

I understand that it lacks a bit of context this equation, but A should be the surjective linear map. I am not sure why:

• $J_A$ depends on $x,y$ if the Jacobian of a linear map is just a constant ? Am I missing something or is it a typo ?

Answer 1

• Yes, the meaning of Euclidean spaces and Euclidean coordinates appear to be what you say and the Lemma is just the singular value decomposition, stated with change of coordinates instead of a product of matrices.
• (1.4) is the general form appearing in the coarea formula. In the case where $A$ is linear, the dependence $J_A(x,y)$ is trivial in the sense that it is constant, nothing mysterious here. The function $f(x)=1$ is a function of $x$, just a constant one.