I have a vector field $\overrightarrow{u}=(x,{z}^{2},y)$ over a tetrahedron with vertices at $(0,0,0),(1,0,0),(0,1,0)$ and $(0,0,2)$ and I need to compute ${\int}_{S}\overrightarrow{u}\cdot d\overrightarrow{S}$ being the surface of the tetrahedron.

I have already calculated ${\int}_{V}(\mathrm{\nabla}\cdot \overrightarrow{u})dV$, which I have found to be equal to $\frac{1}{3}$, now I need to calculate the previous integral and assumedly show that the answers agree (Gauss' Theorem).

So far I have parameterised the surface to say ${\overrightarrow{x}}_{s}=(s,t,2-2s-2t)$

Next I find $d\overrightarrow{S}$ as such:

$d\overrightarrow{S}=(\frac{\mathrm{\partial}\overrightarrow{{x}_{S}}}{\mathrm{\partial}s}\times \frac{\mathrm{\partial}\overrightarrow{{x}_{S}}}{\mathrm{\partial}t})dsdt$

Therefore: $\overrightarrow{u}\cdot d\overrightarrow{S}=10{s}^{2}+16st-16s+8{t}^{2}-15t+8$

But, upon calculating the integral ${\int}_{0}^{1}{\int}_{0}^{1}\overrightarrow{u}\cdot d\overrightarrow{S}dsdt$ I get $\frac{5}{2}$. I'm confident in my first integral, but I don't feel as confident in this one as I feel there may be some theoretical misunderstanding with my parameterisation. Any help is appreciated.

I have already calculated ${\int}_{V}(\mathrm{\nabla}\cdot \overrightarrow{u})dV$, which I have found to be equal to $\frac{1}{3}$, now I need to calculate the previous integral and assumedly show that the answers agree (Gauss' Theorem).

So far I have parameterised the surface to say ${\overrightarrow{x}}_{s}=(s,t,2-2s-2t)$

Next I find $d\overrightarrow{S}$ as such:

$d\overrightarrow{S}=(\frac{\mathrm{\partial}\overrightarrow{{x}_{S}}}{\mathrm{\partial}s}\times \frac{\mathrm{\partial}\overrightarrow{{x}_{S}}}{\mathrm{\partial}t})dsdt$

Therefore: $\overrightarrow{u}\cdot d\overrightarrow{S}=10{s}^{2}+16st-16s+8{t}^{2}-15t+8$

But, upon calculating the integral ${\int}_{0}^{1}{\int}_{0}^{1}\overrightarrow{u}\cdot d\overrightarrow{S}dsdt$ I get $\frac{5}{2}$. I'm confident in my first integral, but I don't feel as confident in this one as I feel there may be some theoretical misunderstanding with my parameterisation. Any help is appreciated.