What is the parametrization of this tetrahedron so that it satisfies assumption in Stokes' theorem?

Question

This question is about a problem in Apostol's Calculus, Vol II, section 12.13 about Stokes' theorem. A question about this problem has been asked before, but that question is about solving the integrals involved in Stokes' theorem. My question is about checking that the assumptions in the theorem are satisfied so that we are allowed to invoke the theorem.

Consider the vector field

$$\pmb{F}(x,y,z)=(xz)\pmb{i}-y\pmb{j}+x^2y\pmb{k}\tag{1}$$

and the surface $S$ consisting of the three faces not in the $xz$-plane of the tetrahedron bounded by the three coordinate planes and the plane $3x+y+3z=6$. The normal $\pmb{n}$ is the unit normal pointing out of the tetrahedron.

Here is a depiction of this surface (all faces of the tetrahedron except for the one on the left on the $xz$-plane)

Suppose we would like to calculate the surface integral of the curl of $\pmb{F}$ through $S$.

My question is about the use of Stokes' theorem to accomplish this.

Stokes' theorem as it appears in Apostol is

1. If we have a smooth parametric surface $S=\pmb{r}(T)$, where $T$ is a region in the $uv$-plane bounded by a piecewise smooth Jordan curve $\Gamma$, and

2. $\pmb{r}$ is a one-to-one mapping whose components have continuous second-order partial derivatives on some open set containing $T\cup\Gamma$, and

3. $C$ is the image of $\Gamma$ under $\pmb{r}$, and

4. $P, Q$, and $R$ are continuously differentiable scalar fields on $S$, then

$$\iint_S \text{curl}(\pmb{F}) dS = \int_C \pmb{F}\cdot d\pmb{r}\tag{2}$$

and here is a nice picture from Apostol's Calculus showing what this all looks like

Now, for the tetrahedron from the start of this post, I think that the curve $C$ is the green curve below

We can calculate the line integral of $\pmb{F}$ along this path.

But my question is about the assumptions in Stokes' theorem.

For the surface in question, what is the parameterization of the surface? That is, what are $T$ and $\pmb{r}$?

It seems that the surface is formed by three different faces with different parameterizations.

I can find the parameterizations of the individual faces and they do each satisfy the assumptions of the theorem, but it's not clear why this means that the theorem applies to the surface as a whole.