Notation and Context (Might not be helpful)
Let $S\subset\mathbb{R}^n$ be a countably $\mathcal{H}^{M}$-rectifiable set. If $f:\mathbb{R}^n\to\mathbb{R}^k$ is a Lipschitz function, then we can find subset $S_+\subset S$ on which $S$ has a tangent space, $\nabla^S f$ exists, and $\nabla^S f \not= 0$. For this subset the following statement holds:
For almost every $t\in\mathbb{R}^k$, $S_t = f^{-1}(t)\cap S_+$ is a countably $\mathcal{H}^{M-k}$-rectifiable set. Moreover, given an $\mathcal{H}^{M}$ measurable function $g$ on $S$, we have the coarea formula
$$\int_{\mathbb{R}^k}\left(\int_{S_t} g\,d\mathcal{H}^{M-k}\right)\,d\mathcal{H}^k = \int_S g\,(J_k^Sf) \,d\mathcal{H}^{M}.$$
Here $J_k^S f$ is the $k$-dimensional Jacobian of $f$ on the surface $S$. For a more detailed discussion, look at section 5.3-5.4, and perhaps section 7.6 of Krantz and Parks' book Geometric Integration Theory, available online at http://www.math.wustl.edu/~sk/books/root.pdf.
Actual Question
My question is about a special case where $f$ does not satisfy the hypotheses above, but is still a fairly nice function. Let $M=2$ and $n=3$, so that $S$ is a two-dimensional surface in $\mathbb{R}^3$. Let $f:\mathbb{R}^3 \to \mathbb{S}^1$ be a map into the 1-sphere $\mathbb{S}^1\subset\mathbb{R}^2$ defined by $$ f(x,y,z) = \frac{(x,y)}{\sqrt{x^2+y^2}}. $$ First off, this is not a Lipschitz function because of its behavior at the origin. The second problem is that I don't want to think of $f$ as a map into $\mathbb{R}^2$, because then both sides of the coarea formula are zero and that's not useful to me.
What I actually expect to be true is:
$$\int_{\mathbb{S}^1}\left(\int_{f^{-1}(\theta)\cap S} \,d\mathcal{H}^{1}\right)\,d\mathcal{H}^1(\theta) = \int_S |x^2+y^2|^{-1/2} \,d\mathcal{H}^{2}.$$
This is the coarea formula written above with $g=1$, $k=1$ (since $f$ maps into a one-dimensional set $\mathbb{S}^1$), and I used $|x^2+y^2|^{-1/2}$ for $J_1^S f$.
In summary, what I want to know is
Is the formula I stated true?
What about higher-dimensional analogues where $f:\mathbb{R}^n\to\mathbb{S}^{n-2}$?