Surface Integrals, Stokes, and the Divergence Theorem曲面积分、斯托克斯定理与散度定理
University-Style Practice Problems大学风格练习题
MEDIUMHARDCOREPROOFAPPLIED
Sections 1 to 7: curl and divergence, parametric surfaces and surface area, scalar surface integrals, flux integrals, Stokes’ Theorem, the Divergence Theorem, and the unified FTC patternCALC III第 1 至 7 节：旋度与散度、参数曲面与曲面面积、标量曲面积分、通量积分、斯托克斯定理、散度定理以及统一的微积分基本定理模式CALC III
Name:姓名：Date:日期：
PART I · CORE TECHNIQUES第一部分 · 核心技巧Computational fluency · 28 marks计算能力 · 28分
Curl, Divergence, and Surface Integrals旋度、散度与曲面积分
Show all working. State which formula you are using at each step (curl determinant, divergence sum, or area element). In flux integrals, check the orientation of your normal before evaluating.展示全部过程。在每一步注明所使用的公式（旋度行列式、散度求和或面积元素）。在通量积分中，计算前请检查法向量的方向。
Q1MEDIUMCOREcurl and divergence of a vector field向量场的旋度与散度[8 marks]
Let $\mathbf{F} = \langle xy^2,\; y z^2,\; x^2 z \rangle$.
(a)Compute $\nabla \times \mathbf{F}$ using the symbolic determinant. State each component explicitly.用符号行列式计算 $\nabla \times \mathbf{F}$，逐分量写出结果。[4]
(c)Evaluate $\nabla \cdot (\nabla \times \mathbf{F})$ without further computation. State the identity you are using and justify why it holds.不需进一步计算，直接求 $\nabla \cdot (\nabla \times \mathbf{F})$。写出所用恒等式并说明其成立的原因。[2]
Consider the surface $S$ given by the graph $z = 2x + 3y$ over the rectangle $D = [0,1]\times[0,1]$, and also the paraboloid $z = x^2 + y^2$ over the disk $x^2 + y^2 \le 1$.考虑以下两个曲面：$D = [0,1]\times[0,1]$ 上的图形曲面 $z = 2x + 3y$，以及圆盘 $x^2 + y^2 \le 1$ 上的抛物面 $z = x^2 + y^2$。
(a)Compute the surface area of the planar piece $z = 2x + 3y$ over $D$. Identify $g_x$ and $g_y$ and simplify the area element $dS$.计算 $D$ 上平面曲面 $z = 2x + 3y$ 的面积，写出 $g_x$ 与 $g_y$，并化简面积元素 $dS$。[3]
(b)Show that the area element for the paraboloid $z = x^2 + y^2$ satisfies $dS = \sqrt{1 + 4r^2}\, dr\, d\theta$ in polar coordinates. Then evaluate $A = \displaystyle\int_0^{2\pi}\!\!\int_0^1 \sqrt{1 + 4r^2}\, r\, dr\, d\theta$. You may use the substitution $u = 1 + 4r^2$.证明抛物面 $z = x^2 + y^2$ 的面积元素在极坐标下满足 $dS = \sqrt{1 + 4r^2}\, dr\, d\theta$，然后计算 $A = \displaystyle\int_0^{2\pi}\!\!\int_0^1 \sqrt{1 + 4r^2}\, r\, dr\, d\theta$，可使用换元 $u = 1 + 4r^2$。[5]
Q3MEDIUMCOREscalar surface integral over a cone锥面上的标量曲面积分[6 marks]
Let $S$ be the cone $z = \sqrt{x^2 + y^2}$, $0 \le z \le 2$, with surface density $\rho(x,y,z) = z^2$. Find the total mass $m = \iint_S z^2\, dS$.设 $S$ 为锥面 $z = \sqrt{x^2 + y^2}$，$0 \le z \le 2$，曲面密度为 $\rho(x,y,z) = z^2$。求总质量 $m = \iint_S z^2\, dS$。
(a)Compute the area element $dS$ for the cone. Show that $\sqrt{1 + g_x^2 + g_y^2} = \sqrt{2}$.计算锥面的面积元素 $dS$，证明 $\sqrt{1 + g_x^2 + g_y^2} = \sqrt{2}$。[2]
(b)Substitute $z = r$ on the cone and set up the integral in polar coordinates over the disk $r \le 2$.在锥面上令 $z = r$，在圆盘 $r \le 2$ 上用极坐标建立积分。[2]
(c)Evaluate the resulting integral to find $m$.计算所得积分，求出 $m$。[2]
Q4HARDCOREflux integral through a graph surface图形曲面的通量积分[6 marks]
Let $\mathbf{F} = \langle y,\; z,\; x \rangle$ and let $S$ be the part of the plane $z = 4 - x - y$ lying over the triangle $D$ with vertices $(0,0)$, $(2,0)$, and $(0,2)$, oriented with upward normal.设 $\mathbf{F} = \langle y,\; z,\; x \rangle$，$S$ 为平面 $z = 4 - x - y$ 位于三角形 $D$（顶点为 $(0,0)$、$(2,0)$、$(0,2)$）上方的部分，取向上的法向量。
(a)Write down the upward vector area element $d\mathbf{S}$ for the graph $z = 4 - x - y$.写出图形曲面 $z = 4 - x - y$ 的向上向量面积元素 $d\mathbf{S}$。[2]
(b)Compute the dot product $\mathbf{F} \cdot d\mathbf{S}$ on the surface, substituting $z = 4 - x - y$. Show that the integrand simplifies to a constant, and state its value.在曲面上代入 $z = 4 - x - y$ 计算点积 $\mathbf{F} \cdot d\mathbf{S}$，证明被积表达式化简为常数，并写出其值。[2]
(c)Evaluate the flux $\iint_S \mathbf{F} \cdot d\mathbf{S}$ by finding the area of $D$.利用 $D$ 的面积计算通量 $\iint_S \mathbf{F} \cdot d\mathbf{S}$。[2]
PART II · DEFINITIONS AND PROOF第二部分 · 定义与证明Rigorous arguments · 26 marks严格论证 · 26分
The Great Theorems重要定理
These items are graded on the logic of the argument. In a theorem proof, verify every hypothesis before invoking the conclusion. In a statement item, produce the correct form of the theorem and explain each piece of notation. Orientation and hypotheses earn method marks separately from the final answer.本部分按论证的逻辑评分。在证明定理时，需在引用结论之前逐一验证所有前提条件。在陈述题中，写出定理的正确形式并解释每个符号的含义。方向与前提条件的得分与最终答案分开计算。
Q5MEDIUMPROOFverifying Stokes’ Theorem on a flat disk在平面圆盘上验证斯托克斯定理[8 marks]
Let $\mathbf{F} = \langle -y^2,\; x,\; z \rangle$. Let $S$ be the disk $x^2 + y^2 \le 4$ in the plane $z = 1$, oriented upward, with boundary the circle $C$: $x^2 + y^2 = 4$, $z = 1$, traversed counterclockwise when viewed from above.设 $\mathbf{F} = \langle -y^2,\; x,\; z \rangle$。设 $S$ 为平面 $z = 1$ 上的圆盘 $x^2 + y^2 \le 4$，取向上的法向量，其边界为圆 $C$：$x^2 + y^2 = 4$，$z = 1$，从上方俯视沿逆时针方向遍历。
(b)Evaluate the curl-flux side of Stokes’ Theorem: $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$.计算斯托克斯定理的旋度通量侧：$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$。[3]
(c)Evaluate the circulation side directly: parametrize $C$ and compute $\oint_C \mathbf{F} \cdot d\mathbf{r}$. Confirm the two sides agree.直接计算环量侧：参数化 $C$ 并计算 $\oint_C \mathbf{F} \cdot d\mathbf{r}$，验证两侧相等。[3]
Q6HARDPROOFthe Divergence Theorem: statement, hypotheses, and proof sketch散度定理：陈述、前提条件与证明思路[10 marks]
This question asks you to state and work with the Divergence Theorem (Gauss’ Theorem) carefully.本题要求你仔细陈述并运用散度定理（高斯定理）。
(a)State the Divergence Theorem precisely, including the hypotheses on the region $E$, the surface $\partial E$, and the field $\mathbf{F}$. Use correct notation.精确陈述散度定理，包括对区域 $E$、曲面 $\partial E$ 和向量场 $\mathbf{F}$ 的前提条件，使用正确的符号。[3]
(b)Let $\mathbf{F} = \langle x,\; y^2,\; z^3 \rangle$ and let $E$ be the solid ball $x^2 + y^2 + z^2 \le a^2$. Use the Divergence Theorem to find the outward flux through the sphere $\partial E$ in terms of $a$. State $\nabla \cdot \mathbf{F}$ and set up the triple integral in spherical coordinates before evaluating. You may use $\iiint_E 1\,dV = \tfrac{4}{3}\pi a^3$, $\iiint_E z^2\,dV = \tfrac{4}{15}\pi a^5$, and note by symmetry that $\iiint_E y\,dV = 0$.设 $\mathbf{F} = \langle x,\; y^2,\; z^3 \rangle$，$E$ 为实心球 $x^2 + y^2 + z^2 \le a^2$。用散度定理求通过球面 $\partial E$ 的向外通量（以 $a$ 表示）。写出 $\nabla \cdot \mathbf{F}$，在计算之前用球坐标建立三重积分。可使用 $\iiint_E 1\,dV = \tfrac{4}{3}\pi a^3$、$\iiint_E z^2\,dV = \tfrac{4}{15}\pi a^5$，并由对称性知 $\iiint_E y\,dV = 0$。[5]
(c)Explain in two to three sentences why the Divergence Theorem fails for the field $\mathbf{G} = \dfrac{\langle x, y, z\rangle}{(x^2 + y^2 + z^2)^{3/2}}$ when $E$ is the unit ball. What is $\nabla \cdot \mathbf{G}$ away from the origin, and why does this not force the flux to be zero?用两到三句话解释为何当 $E$ 为单位球时散度定理对向量场 $\mathbf{G} = \dfrac{\langle x, y, z\rangle}{(x^2 + y^2 + z^2)^{3/2}}$ 失效。在原点以外 $\nabla \cdot \mathbf{G}$ 等于多少？为什么这不能迫使通量为零？[2]
Q7HARDPROOFthe four theorems as a unified FTC pattern四大定理作为统一的微积分基本定理模式[8 marks]
This question concerns the dimensional ladder of the four boundary theorems.本题涉及四个边界定理的维度阶梯。
(a)Write down all four theorems (the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem) in a single aligned display, labelling which dimension each acts on and which derivative ($\nabla f$, $Q_x - P_y$, $\nabla \times \mathbf{F}$, or $\nabla \cdot \mathbf{F}$) appears.在一个对齐展示中写出全部四个定理（线积分基本定理、格林定理、斯托克斯定理和散度定理），标注每个定理作用的维度以及出现的导数（$\nabla f$、$Q_x - P_y$、$\nabla \times \mathbf{F}$ 或 $\nabla \cdot \mathbf{F}$）。[4]
(b)Two identities link the operators: $\nabla \times (\nabla f) = \mathbf{0}$ and $\nabla \cdot (\nabla \times \mathbf{F}) = 0$. Prove the second identity by expanding $\nabla \cdot (\nabla \times \mathbf{F})$ in terms of partial derivatives and invoking Clairaut’s Theorem.两个恒等式联系着这些算子：$\nabla \times (\nabla f) = \mathbf{0}$ 和 $\nabla \cdot (\nabla \times \mathbf{F}) = 0$。通过将 $\nabla \cdot (\nabla \times \mathbf{F})$ 展开为偏导数并引用克莱罗定理来证明第二个恒等式。[4]
PART III · APPLICATIONS AND SYNTHESIS第三部分 · 应用与综合Extended problems · 28 marks综合题 · 28分
Theorem Strategy and Flux Computation定理策略与通量计算
Set up each problem cleanly before integrating. State which theorem you are applying and verify the required hypotheses (closed surface for Divergence Theorem; consistent right-hand orientation for Stokes). Where two methods are available, use the easier one and note why it is easier.在积分之前清晰地建立每道题的框架。说明所使用的定理，并验证所需前提条件（散度定理要求封闭曲面；斯托克斯定理要求右手定则方向一致）。若有两种方法可用，选择较简单的一种并说明原因。
Q8HARDAPPLIEDStokes replacing a hard surface integral by a line integral斯托克斯定理将复杂曲面积分转化为线积分[10 marks]
Let $\mathbf{F} = \langle z^2 - y,\; z^2 + x,\; 0 \rangle$ and let $S$ be the upper hemisphere $x^2 + y^2 + z^2 = 4$, $z \ge 0$, oriented with outward (upward) normal. Let $C$ be the boundary circle $x^2 + y^2 = 4$, $z = 0$, with counterclockwise orientation when viewed from above.设 $\mathbf{F} = \langle z^2 - y,\; z^2 + x,\; 0 \rangle$，$S$ 为上半球面 $x^2 + y^2 + z^2 = 4$，$z \ge 0$，取向外（向上）法向量。设 $C$ 为边界圆 $x^2 + y^2 = 4$，$z = 0$，从上方俯视沿逆时针方向。
(a)Compute $\nabla \times \mathbf{F}$. Write all three components.计算 $\nabla \times \mathbf{F}$，写出全部三个分量。[3]
(b)Explain why computing $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ directly over the hemisphere is difficult. Identify a simpler surface $D$ sharing the same oriented boundary $C$, and state the right-hand rule condition that makes the substitution valid.解释为何直接在半球面上计算 $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ 较为困难。指出一个与 $C$ 共享同一有向边界的更简单曲面 $D$，并说明使替换合法的右手定则条件。[2]
(c)On $z = 0$ the field simplifies to $\mathbf{F} = \langle -y,\; x,\; 0\rangle$. Parametrize $C$ by $\mathbf{r}(t) = \langle 2\cos t,\; 2\sin t,\; 0\rangle$ and evaluate $\oint_C \mathbf{F} \cdot d\mathbf{r}$. State the conclusion for $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$.在 $z = 0$ 处向量场化简为 $\mathbf{F} = \langle -y,\; x,\; 0\rangle$。用 $\mathbf{r}(t) = \langle 2\cos t,\; 2\sin t,\; 0\rangle$ 参数化 $C$ 并计算 $\oint_C \mathbf{F} \cdot d\mathbf{r}$，写出 $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ 的结论。[5]
Q9HARDAPPLIEDDivergence Theorem for a closed surface, then open surface by subtraction封闭曲面上的散度定理，然后用减法求开放曲面通量[10 marks]
Let $\mathbf{F} = \langle xz,\; yz,\; z^2 \rangle$. Let $E$ be the solid region bounded below by the paraboloid $z = x^2 + y^2$ and above by the plane $z = 1$. Let $S_{\text{par}}$ be the paraboloid cap $z = x^2 + y^2$, $0 \le z \le 1$, and let $T$ be the top disk $z = 1$, $x^2 + y^2 \le 1$.设 $\mathbf{F} = \langle xz,\; yz,\; z^2 \rangle$。设 $E$ 为下方由抛物面 $z = x^2 + y^2$、上方由平面 $z = 1$ 围成的实心区域。设 $S_{\text{par}}$ 为抛物面帽 $z = x^2 + y^2$，$0 \le z \le 1$，$T$ 为顶部圆盘 $z = 1$，$x^2 + y^2 \le 1$。
(a)Compute $\nabla \cdot \mathbf{F}$ and evaluate $\displaystyle\iiint_E (\nabla \cdot \mathbf{F})\, dV$ using cylindrical coordinates. This gives the total outward flux through the closed surface $\partial E = T \cup S_{\text{par, out}}$.计算 $\nabla \cdot \mathbf{F}$，并用柱坐标计算 $\displaystyle\iiint_E (\nabla \cdot \mathbf{F})\, dV$，这给出通过封闭曲面 $\partial E = T \cup S_{\text{par, out}}$ 的总向外通量。[5]
(b)Compute the outward flux through the top disk $T$ (outward normal $\mathbf{n} = \langle 0, 0, 1\rangle$, $z = 1$ on $T$).计算通过顶部圆盘 $T$ 的向外通量（向外法向量 $\mathbf{n} = \langle 0, 0, 1\rangle$，$T$ 上 $z = 1$）。[2]
(c)Hence find the outward flux of $\mathbf{F}$ through the paraboloid $S_{\text{par}}$ with outward-from-$E$ (downward) orientation. Then state the flux with the upward orientation that a student would typically describe as "through the bowl looking upward."从而求 $\mathbf{F}$ 通过抛物面 $S_{\text{par}}$（取从 $E$ 向外即向下的方向）的通量。然后写出取向上方向（即"从碗底向上看"方向）时的通量。[3]
Q10HARDAPPLIEDdirect flux vs Divergence Theorem: comparing two methods直接通量与散度定理：比较两种方法[8 marks]
Let $\mathbf{F} = \langle x^3,\; y^3,\; z^3 \rangle$ and let $S$ be the closed surface of the cylinder $x^2 + y^2 \le 1$, $0 \le z \le 2$, with outward orientation.设 $\mathbf{F} = \langle x^3,\; y^3,\; z^3 \rangle$，$S$ 为圆柱体 $x^2 + y^2 \le 1$，$0 \le z \le 2$ 的封闭曲面，取向外法向量。
(a)Compute the outward flux via the Divergence Theorem. State $\nabla \cdot \mathbf{F}$ and evaluate the resulting triple integral in cylindrical coordinates.用散度定理计算向外通量。写出 $\nabla \cdot \mathbf{F}$ 并用柱坐标计算所得三重积分。[4]
(b)Verify the answer by computing the flux directly: find the flux through the curved lateral surface, the top disk ($z = 2$), and the bottom disk ($z = 0$) separately, then sum them. Show your normal vectors.通过直接计算验证答案：分别求通过弯曲侧面、顶部圆盘（$z = 2$）和底部圆盘（$z = 0$）的通量，然后求和。写出法向量。[4]