I have to parameterise the faces of the tetrahedron, $z = 0$, $y=0$, $x=y$, $x+z=1$ and use their normal vectors to find the surface integral $\int_Sxy\ dS$. I'm not sure if I have parameterised correctly and especially unsure about the surface integral.
My attempt:
$z = 0:$
$x\mathbf{e_1} + y\mathbf{e_2}$
Normal vector: $
\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bmatrix}
$
$y = 0:$
$x\mathbf{e_1} + z\mathbf{e_3}$
Normal vector: $
\begin{bmatrix}
0 \\
1 \\
0 \\
\end{bmatrix}$
$x = y:$
$y\mathbf{e_1} + y\mathbf{e_2} + z\mathbf{e_3}$
Normal vector: $
\begin{bmatrix}
1 \\
-1 \\
0 \\
\end{bmatrix}$
$x + z = 1:$
$x\mathbf{e_1} + y\mathbf{e_2} + (1-x)\mathbf{e_3}$
Normal vector: $
\begin{bmatrix}
1 \\
0 \\
1 \\
\end{bmatrix}$
Then my method was to integrate over each face at a time and add them together but using the parameterisation so they were all double integrals and using the modulus of the normal vectors as the change of variables bit.
Eg:
For $z = 0$:
$\int_0^1\int_0^1xy\ dx\ dy = ... = \frac{1}{4}$