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1.  Evaluate the flux integral \(\int\int_S \langle x, y, z\rangle \cdot \vec{N}dS\) where \(S\) is the surface \(z = 2 - x - y\) where \(0 \leq x \leq 1, 0 \leq y \leq 1\).

a.  -2

b.  -1

c.  0

d.  1

e.  2

2.  Evaluate the surface integral \(\int\int_S \langle 3z, -4, y\rangle \cdot \vec{N}dS\) where \(S: x + y + z = 1\), \(x \geq 0\), \(y \geq 0\), \(z \geq 0\). (As is typical in this situation, S has the upward orientation.)

a.  \(-\frac{1}{3}\)

b.  \(-\frac{2}{3}\)

c.  -1

d.  \(-\frac{4}{3}\)

e.  \(-\frac{5}{3}\)

3.  Evaluate the surface integral \(\int\int_S \langle x, y, z\rangle \cdot \vec{N}dS\) where \(S\) is the unit sphere centered at the point (1, 2, 5).(As is typical in this situation, S has the outward orientation.)

a.  \(2\pi\)

b.  \(3\pi\)

c.  \(4\pi\)

d.  \(5\pi\)

e.  \(6\pi\)

4.  Say \(\vec{F}\) is the velocity field for the flow of any incompressible fluid (like liquid water), and S is any bounded closed surface with the outward orientation. Then what can we say about \(\int\int_S \vec{F} \cdot \vec{N}dS\)?

a.  It's 0 because the divergence is 0 everywhere.

b.  It's 0 because the curl is 0 everywhere.

c.  It could be any value because all real numbers should be respected.

d.  It's never 0 because it's never really over.

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