Relationship between surface integral and fubini's theorem

Question

I am studying about surface integral and recently have studied about fubini's theorem.

And I think there exists any relationship between 'surface integral' and 'fubini's theorem'.

As a result, I came to think that there would be a relationship between the two, but I'm not confident: may be it is wrong or more restrictive.

The process of deriving a relationship is as follows:

I'll calculate $\int\int _{S} ^{} {fdS}$ and the process calcuate surface integral of $f$ around surface $S$ is:

1. Parameterize $S$ by defining a system of curvilinear coordinate on S.

2. Let such a parameterization be $\vec{r}(s,t)$, where (s,t) varies in some region T in the plane.

3.Then, the surface integral is given by $\int\int _{S} ^{} {fdS} = \int\int _{T} ^{}\ {f( \vec{{r}} (s,t))\vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t}}\vert\vert dsdt$.

In here

1. let $S=H \times L$ in curvilinear coordinate $(h,l,u)$ (Since curvilinear coordinate defined on S, $u$ is constant)
2. Let $T=H \times L$ in coordinate coordinate $(s,t,z)$.
3. Suppose that $\vec{r}(s,t): T\rightarrow S $ defined by $\vec{r}:(s,t)\rightarrow(s,t,u)$.

I think 2,3 may be reasonable if 1 is reasonable, becuase curvilinear coordinates are derived from a set of cartesian coordinates by using a transformation that is invertible at each point.

Then, by fubini's theorem, $\int\int _{S} ^{} {fdS} = \int\int _{T} ^{}\ {f( \vec{{r}} (s,t))\vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t}}\vert\vert dsdt$=$\int_{L}\int_{H}f(s,t) \vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t} \vert\vert dsdt $.

But also, according to the fubini's theorem, $\int\int _{S} ^{} {fdS} =\int_L\int_H f(h,l)dhdl$.

Then, the above two equations yield the following: $\int_{L}\int_{H}f(s,t) \vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t} \vert\vert dsdt=\int\int _{S} ^{} {fdS} =\int_L\int_H f(h,l)dhdl$, which implies that $ \vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t} \vert\vert=1$.

Thus, I have conclude that ,for a curvilinear coordinate defined on surface $S$, if $S=H \times L$, there exist parametric equation $\vec{r}$ such that $ \vert\vert \frac {\partial \vec{r}}{\partial s} \times \frac{\partial{\vec{r}}}{\partial t} \vert\vert=1$ in cartesian coordinate (s,t,z).

What was wrong??