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Tutorial Exercise
Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ for the given vector field $\mathbf{F}$ and the oriented surface $S$. In other words, find the flux of $\mathbf{F}$ across $S$. For closed surfaces, use the positive (outward) orientation
$\mathbf{F}(x, y, z) = xze^x \mathbf{i} - xze^x \mathbf{j} + z \mathbf{k}$
$S$ is the part of the plane $x + y + z = 8$ in the first octant and has downward orientation.
Step 1
Since $S$ is a portion of the graph of $z = g(x, y) = 8 - x - y$, then
$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \left( -P \frac{\partial g}{\partial x} - Q \frac{\partial g}{\partial y} + R \right) dA$.
Therefore, since $S$ has downward orientation, we have
$\iint_S \mathbf{F} \cdot d\mathbf{S} = - \iint_D \left( \boxed{ } - y \right) dA.$
