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This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. 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 $\iint_S \mathbf{F} \cdot d\mathbf{S} = - \iint_D \left( \boxed{ } - y \right) dA.$

          This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.
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
$\iint_S \mathbf{F} \cdot d\mathbf{S} = - \iint_D \left( \boxed{ } - y \right) dA.$

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
Evaluate the surface integral 𝐅· d𝐒 for the given vector field 𝐅 and the oriented surface S. In other words, find the flux of 𝐅 across S. For closed surfaces, use the positive (outward) orientation
𝐅(x, y, z) = xze^x 𝐢 - xze^x 𝐣 + z 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
𝐅· d𝐒 = ( -P (∂ g)/(∂ x) - Q (∂ g)/(∂ y) + R ) dA.
Therefore, since S has downward orientation, we have
𝐅· d𝐒 = - ( - y ) dA.

Added by Cynthia B.

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