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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) = x² + 2y². Which of the following must be true? (A) f(x, y) is concave upward on the line $y = -x (B) the relative minimum of f(x, y) subject to the constraint x + y = 1 is (C) the absolute minimum of f(x, y) on its domain does not exist (D) the minimum rate of change of f(x, y) at the point P(1, -1) is
3. Which of the following must be true? (A) The series converges for all p≠0$ (B) The series converges absolutely (C) The series converges (D) The series diverges
2. Which of the following must be true? (A) (B) (C) (D)

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