Chapter 37 — Exercises

39 problems on Stokes' theorem, the Divergence theorem, choosing the easier side, and Maxwell/physics applications. Tiered ⭐ to ⭐⭐⭐⭐.

Work these by hand unless a problem explicitly invites Python. Fix every orientation with the right-hand rule before you compute, and keep the governing slogan of §37.1 in view: the integral of a derivative over a region equals an integral over its boundary. When a flux through a closed surface or a circulation around a loop looks ugly, ask which side of the theorem is easier — choosing the friendly side (§37.2, §37.5) is half the skill.

Curl and divergence are computed exactly as in Chapter 34; Green's theorem (the 2D case) is Chapter 35; surface integrals and flux are Chapter 36; the original FTC is Chapter 14.

Difficulty tiers: ⭐ recall/direct computation · ⭐⭐ standard application · ⭐⭐⭐ multi-step / requires a choice · ⭐⭐⭐⭐ synthesis, proof, or modeling.

Part A — Curl, Divergence, and Setup (⭐)

A1. ⭐ Compute $\nabla\times\mathbf{F}$ for $\mathbf{F} = \langle -y,\, x,\, 0\rangle$.

A2. ⭐ Compute $\nabla\cdot\mathbf{F}$ for $\mathbf{F} = \langle x,\, y,\, z\rangle$.

A3. ⭐ Compute $\nabla\cdot\mathbf{F}$ for $\mathbf{F} = \langle x^2,\, y^2,\, z^2\rangle$.

A4. ⭐ Compute $\nabla\times\mathbf{F}$ for $\mathbf{F} = \langle z,\, x,\, y\rangle$.

A5. ⭐ State, from memory, both Stokes' theorem and the Divergence theorem, naming the region, the boundary, and the "derivative" appearing on each side. Identify which side is a boundary integral and which is a region integral (§37.6).

A6. ⭐ For the unit disk in the $xy$-plane oriented with normal $+\hat{\mathbf{z}}$, which way (clockwise or counterclockwise viewed from above) must the boundary circle be traversed for Stokes' theorem to hold? Justify with the right-hand rule (§37.2).

A7. ⭐ Verify the identity $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ for $\mathbf{F} = \langle yz,\, xz,\, xy\rangle$ by direct computation (this is the identity behind surface-independence, §37.9).

Part B — Verifying Stokes' Theorem (⭐⭐)

B1. ⭐⭐ Verify Stokes' theorem for $\mathbf{F} = \langle -y,\, x,\, 0\rangle$ over the disk $x^2 + y^2 \le 4$, $z = 0$, oriented upward. Compute both sides and confirm they agree.

B2. ⭐⭐ Let $\mathbf{F} = \langle z,\, x,\, y\rangle$ and let $S$ be any surface whose boundary is the unit circle in the $xy$-plane (normal up). Use Stokes' theorem to find $\oint_{\partial S}\mathbf{F}\cdot d\mathbf{r}$ without parametrizing the circle.

B3. ⭐⭐ Verify Stokes' theorem for $\mathbf{F} = \langle 2y,\, 3z,\, x\rangle$ over the triangular region with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ in the $xy$-plane, oriented upward. (Hint: the curl-flux side is a constant times the triangle's area.)

B4. ⭐⭐ For $\mathbf{F} = \langle y,\, -x,\, 0\rangle$, compute the circulation around the unit circle (CCW, normal up) two ways: directly as a line integral and via Stokes' theorem. Why is the result negative?

B5. ⭐⭐ Let $\mathbf{F} = \langle x^2,\, 2x,\, z^2\rangle$. Use Stokes' theorem to evaluate $\oint_C \mathbf{F}\cdot d\mathbf{r}$ where $C$ is the boundary of the disk $x^2 + y^2 \le 1$ in the plane $z = 3$, oriented counterclockwise viewed from above.

B6. ⭐⭐ A field has $\nabla\times\mathbf{F} = \langle 0,0,0\rangle$ everywhere on $\mathbb{R}^3$. Without knowing $\mathbf{F}$, what is $\oint_C\mathbf{F}\cdot d\mathbf{r}$ around any closed curve $C$? Name the property of $\mathbf{F}$ (§37.9, Chapter 35).

Part C — Surface Independence and Choosing the Easier Side (⭐⭐⭐)

C1. ⭐⭐⭐ Let $\mathbf{F} = \langle -y,\, x,\, 0\rangle$. Compute the curl-flux through the upper hemisphere $x^2+y^2+z^2 = 1$, $z\ge 0$ (outward normal) by replacing the hemisphere with the flat unit disk that shares its boundary. State the principle that licenses the swap (§37.2, §37.9).

C2. ⭐⭐⭐ Let $\mathbf{F} = \langle 2z,\, x^2,\, -y\rangle$ and let $S$ be the part of the paraboloid $z = 4 - x^2 - y^2$ above the $xy$-plane, oriented upward. Evaluate $\iint_S(\nabla\times\mathbf{F})\cdot d\mathbf{S}$ by converting it to a line integral around the boundary circle $x^2+y^2 = 4$, $z = 0$.

C3. ⭐⭐⭐ You must compute $\oint_C \mathbf{F}\cdot d\mathbf{r}$ for a closed curve $C$ bounding an awkward surface, with $\mathbf{F} = \langle e^{x}+y,\, \sin y - z,\, x + z^2\rangle$. Decide which side of Stokes' theorem to use, compute $\nabla\times\mathbf{F}$, and explain in one sentence why the curl side might be easier here.

C4. ⭐⭐⭐ Two surfaces $S_1$ (a flat disk) and $S_2$ (a tall cone) share the same boundary circle. Without computing, argue from §37.9 why $\iint_{S_1}(\nabla\times\mathbf{F})\cdot d\mathbf{S} = \iint_{S_2}(\nabla\times\mathbf{F})\cdot d\mathbf{S}$. Which identity — $\nabla\cdot(\nabla\times\mathbf{F})=0$ or $\nabla\times(\nabla f)=\mathbf{0}$ — does the argument use?

C5. ⭐⭐⭐ Let $\mathbf{F} = \langle xy,\, yz,\, zx\rangle$ and let $S$ be the part of the plane $x + y + z = 1$ in the first octant, oriented with upward-pointing normal. Set up and evaluate $\iint_S(\nabla\times\mathbf{F})\cdot d\mathbf{S}$ by projecting onto the $xy$-plane.

Part D — Verifying the Divergence Theorem (⭐⭐)

D1. ⭐⭐ Verify the Divergence theorem for $\mathbf{F} = \langle x,\, y,\, z\rangle$ over the ball $x^2+y^2+z^2 \le 4$. Compute both the volume integral of the divergence and the flux through the sphere.

D2. ⭐⭐ Use the Divergence theorem to find the outward flux of $\mathbf{F} = \langle x^2,\, y^2,\, z^2\rangle$ through the surface of the box $[0,1]\times[0,2]\times[0,3]$.

D3. ⭐⭐ Find the outward flux of $\mathbf{F} = \langle 2x,\, 3y,\, z\rangle$ through any closed surface enclosing a solid of volume $V$. (Hint: the divergence is constant.)

D4. ⭐⭐ Verify the Divergence theorem for $\mathbf{F} = \langle x,\, y,\, z\rangle$ over the cube $[0,1]^3$ by computing both sides. Note you must sum six face fluxes for the surface side.

D5. ⭐⭐ Use the Divergence theorem to evaluate the flux of $\mathbf{F} = \langle x^3,\, y^3,\, z^3\rangle$ through the unit sphere. (Convert to spherical coordinates, Chapter 33.)

Part E — Choosing the Easier Side / Closed Surfaces (⭐⭐⭐)

E1. ⭐⭐⭐ Find the outward flux of $\mathbf{F} = \langle xz,\, yz,\, z^2\rangle$ through the closed surface of the cylinder $x^2 + y^2 \le 1$, $0\le z\le 2$ (including top and bottom). Use the Divergence theorem and cylindrical coordinates.

E2. ⭐⭐⭐ The surface $S$ is the open paraboloid bowl $z = x^2 + y^2$, $0\le z\le 1$, oriented with outward normal, and $\mathbf{F} = \langle x,\, y,\, z\rangle$. The Divergence theorem does not apply directly because $S$ is not closed. Cap $S$ with the disk $z = 1$, apply the theorem to the closed solid, then subtract the cap's flux to recover the flux through the bowl alone (§37.5 pitfall).

E3. ⭐⭐⭐ Find the outward flux of $\mathbf{F} = \langle x + \sin(yz),\, y + e^{xz},\, z + xy\rangle$ through the sphere $x^2+y^2+z^2 = 9$. (Hint: most of the divergence is trivial; what is $\nabla\cdot\mathbf{F}$?)

E4. ⭐⭐⭐ A field $\mathbf{F}$ satisfies $\nabla\cdot\mathbf{F} = 0$ everywhere. What is its flux through any closed surface? Name the property (solenoidal, §37.9) and state what kind of potential $\mathbf{F}$ must therefore possess.

E5. ⭐⭐⭐ Compute the flux of $\mathbf{F} = \langle y^2 z,\, xz,\, x^2 y^2\rangle$ outward through the closed surface bounding the solid hemisphere $x^2+y^2+z^2\le 1$, $z\ge 0$. (Find $\nabla\cdot\mathbf{F}$ first; it may simplify dramatically.)

Part F — Physics and Maxwell Applications (⭐⭐⭐)

F1. ⭐⭐⭐ (Gauss's law.) A point charge $q$ sits at the origin, producing $\mathbf{E} = \dfrac{q}{4\pi\varepsilon_0}\dfrac{\mathbf{r}}{\|\mathbf{r}\|^3}$. Compute the flux $\oiint_S \mathbf{E}\cdot d\mathbf{S}$ through a sphere of radius $R$ centered at the origin, and confirm it equals $q/\varepsilon_0$. Why does the Divergence theorem not apply to the solid ball here (§37.5 pitfall)?

F2. ⭐⭐⭐ (Gauss's law for magnetism.) Explain, using the Divergence theorem and the identity $\nabla\cdot(\nabla\times\mathbf{A}) = 0$, why the net magnetic flux $\oiint_S\mathbf{B}\cdot d\mathbf{S}$ through any closed surface is zero when $\mathbf{B} = \nabla\times\mathbf{A}$ (§37.7).

F3. ⭐⭐⭐ (Faraday's law.) Starting from the integral form $\oint_{\partial S}\mathbf{E}\cdot d\mathbf{r} = -\dfrac{d}{dt}\iint_S\mathbf{B}\cdot d\mathbf{S}$, use Stokes' theorem to derive the differential form $\nabla\times\mathbf{E} = -\partial_t\mathbf{B}$. State where you used "true for every surface $S$" (§37.7).

F4. ⭐⭐⭐ (Continuity equation.) Starting from $\dfrac{d}{dt}\iiint_V\rho\,dV = -\oiint_{\partial V}\rho\mathbf{u}\cdot d\mathbf{S}$, derive $\partial_t\rho + \nabla\cdot(\rho\mathbf{u}) = 0$ using the Divergence theorem (§37.8).

Part G — Synthesis, Proof, and Modeling (⭐⭐⭐⭐)

G1. ⭐⭐⭐⭐ (Volume from the boundary.) Prove that for any solid $E$ with boundary $\partial E$, $$\text{Vol}(E) = \frac{1}{3}\oiint_{\partial E}\langle x, y, z\rangle\cdot d\mathbf{S}.$$ Identify the field's divergence, then apply the Divergence theorem. Use the formula to recompute the volume of the unit ball and confirm $\tfrac{4}{3}\pi$.

G2. ⭐⭐⭐⭐ (Why Stokes contains Green.) Show that when $S$ is a flat region in the $xy$-plane with normal $+\hat{\mathbf{z}}$ and $\mathbf{F} = \langle P, Q, 0\rangle$, Stokes' theorem reduces exactly to Green's theorem $\oint P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA$ (Chapter 35). Carry out the curl computation explicitly.

G3. ⭐⭐⭐⭐ (Surface-independence, proved.) Let $S_1$ and $S_2$ share a boundary curve $C$ and together bound a solid $E$. Using the Divergence theorem on the curl field $\nabla\times\mathbf{F}$ and the identity $\nabla\cdot(\nabla\times\mathbf{F}) = 0$, prove that $\iint_{S_1}(\nabla\times\mathbf{F})\cdot d\mathbf{S} = \iint_{S_2}(\nabla\times\mathbf{F})\cdot d\mathbf{S}$ (mind the orientations). This is the rigorous version of §37.2's claim.

G4. ⭐⭐⭐⭐ (Divergence as flux density.) Starting from $$\nabla\cdot\mathbf{F}(P) = \lim_{V\to 0}\frac{1}{V}\oiint_{\partial V}\mathbf{F}\cdot d\mathbf{S},$$ explain how the macroscopic Divergence theorem (§37.5) is just this microscopic identity summed over a partition into tiny cells. Connect to the finite-volume method (§37.6, §37.11).

G5. ⭐⭐⭐⭐ (Applied — physics track.) A spherical region of radius $R$ contains uniform charge density $\rho_0$. Using Gauss's law in integral form and spherical symmetry, derive the electric field magnitude $E(r)$ both inside ($r < R$) and outside ($r > R$) the charge. Verify your inside result satisfies $\nabla\cdot\mathbf{E} = \rho_0/\varepsilon_0$ in differential form (§37.7).

G6. ⭐⭐⭐⭐ (Applied — data-science / engineering track.) A finite-volume solver stores a field's values on the faces of a cubical grid cell. Explain, using the Divergence theorem, why the discrete scheme "net flux out of a cell = sum of source terms inside" conserves the quantity exactly (not approximately) as flux leaves one cell and enters its neighbor (§37.6, §37.11). Then describe a numerical check you would run, with hand-computed expected output (do not execute code).

G7. ⭐⭐⭐⭐ (Heat equation derivation.) From the energy balance $\dfrac{d}{dt}\iiint_V c\rho\,T\,dV = \oiint_{\partial V} k\nabla T\cdot d\mathbf{S}$ and Fourier's law, derive the heat equation $\partial_t T = \alpha\nabla^2 T$ with $\alpha = k/(c\rho)$ via the Divergence theorem (§37.8). Identify the step that requires "true for every region $V$."

Tier Count Summary

Solutions to odd-numbered problems appear in appendices/answers-to-selected.md. If a problem seems to demand a brute-force surface or volume integral, pause: Stokes' theorem and the Divergence theorem exist precisely so you rarely have to (§37.2, §37.5).