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3. Suppose that $f$ and $g$ are scalar functions with continuous first- and second-order partial derivatives throughout a region $E$ that is bounded by a closed piecewise smooth surface $S$. Show that: - $\iint_S f \nabla g \cdot \mathbf{n} dS = \iiint_E (\nabla f \cdot \nabla g + f \nabla^2 g) dV$ - $\iint_S (f \nabla g - g \nabla f) \cdot \mathbf{n} dS = \iiint_E (f \nabla^2 g - g \nabla^2 f) dV$
2. Use the surface integral in Stokes' Theorem to calculate the flux of the field $\mathbf{F} = (x - y)\mathbf{i} + (yz)\mathbf{j} + (z - x)\mathbf{k}$ across the surface $S: \mathbf{r}(\phi, \theta) = (2\sin{\phi}\cos{\theta})\mathbf{i} + (2\sin{\phi}\sin{\theta})\mathbf{j} + 2\cos{\phi}\mathbf{k}, 0 \leq \phi \leq \frac{\pi}{2}, 0 \leq \theta \leq 2\pi$ in the direction of the outward unit normal $\mathbf{n}$.
(b) (5 分) Use the above formula to evaluate $\int_0^{\frac{\pi}{2}} x \sin^5{x} dx$.
4. Set $f(x, y) = x^2 + 2y^2$. Which of the following must be true? (A) $f(x, y)$ is concave upward on the line $y = -x$ (B) The relative minimum of $f(x, y)$ subject to the constraint $x + y = 1$ is $\frac{2}{3}$ (C) The absolute minimum of $f(x, y)$ on its domain does not exist (D) The minimum rate of change of $f(x, y)$ at the point $P(1, -1)$ is $-4\sqrt{5}$
3. Which of the following must be true? (A) The series $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^p}$ converges for all $p \neq 0$ (B) The series $\sum_{k=1}^\infty (-1)^{k-1} k^2 e^{-k}$ converges absolutely (C) The series $\sum_{k=1}^\infty \frac{\cos{(2k)}}{k^2}$ converges (D) The series $\sum_{k=1}^\infty (1 + \frac{1}{k})^k$ diverges
2. Which of the following must be true? (A) $\int_1^4 e^{-x^2} dx = 3$ (B) $\int_1^4 |\frac{x^3}{x^4 + 1}| dx = 3 - e^{-4}$ (C) $\int_2^\infty \frac{1}{\sqrt{x}} dx$ diverges (D) $\int_2^\infty \frac{\sin^2{x}}{x^2} dx$ converges

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