Chapter 34 — Exercises

38 problems on vector fields, streamlines, divergence, curl, conservative fields, and applications. ⭐ to ⭐⭐⭐⭐.

Work these by hand unless a problem explicitly calls for Python. Build the picture first, then the algebra — the two should always agree. Answers to selected problems appear in the back-of-book solutions.

Difficulty tiers: ⭐ foundational · ⭐⭐ standard · ⭐⭐⭐ challenging · ⭐⭐⭐⭐ advanced/synthesis.

Part A — Sketching and Identifying Vector Fields (§34.1–34.2)

1. ⭐ For the constant field $\mathbf{F}(x,y) = \langle 0, -2\rangle$, evaluate the field at $(1,1)$, $(-1,0)$, and $(2,3)$. In one sentence, describe the picture.

2. ⭐ Evaluate the radial field $\mathbf{F}(x,y) = \langle x, y\rangle$ at the four points $(1,0)$, $(0,1)$, $(-1,0)$, $(0,-1)$. Do the arrows point toward or away from the origin?

3. ⭐ Evaluate the rotational field $\mathbf{F}(x,y) = \langle -y, x\rangle$ at $(1,0)$, $(0,1)$, $(-1,0)$, $(0,-1)$. State the direction of circulation (clockwise or counterclockwise).

4. ⭐ Explain in one or two sentences why a vector field $\mathbf{F}(x,y)$ is a different kind of object from the vector-valued function $\mathbf{r}(t)$ of Chapter 28. Identify the input and output of each.

5. ⭐ A student writes a planar vector field as $\mathbf{F}(t)$. What is wrong with this notation, and how should it be written? (See the Common Pitfall in §34.1.)

6. ⭐ Match each field to its description — source, swirl, or uniform: (a) $\langle 3, 1\rangle$; (b) $\langle x, y\rangle$; (c) $\langle -y, x\rangle$; (d) $\langle -x, -y\rangle$.

7. ⭐ The gradient of $f(x,y) = x^2 + y^2$ is a vector field. Compute $\nabla f$ and describe its picture. (Recall the gradient from Chapter 30.)

8. ⭐ For the inverse-square field $\mathbf{F}(x,y) = -\dfrac{1}{r^3}\langle x,y\rangle$ with $r = \sqrt{x^2+y^2}$, find $\|\mathbf{F}\|$ as a function of $r$. Do the arrows point inward or outward?

9. ⭐ Sketch (or describe point-by-point at $(\pm1,0),(0,\pm1)$) the field $\mathbf{F}(x,y) = \langle y, x\rangle$. Is it a source, a swirl, or a saddle-type pattern?

Part B — Streamlines / Flow Lines (§34.3)

10. ⭐⭐ Write the streamline equation $\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$ as a system of two scalar ODEs for the field $\mathbf{F}(x,y) = \langle 2, 3\rangle$, then solve it with initial point $(x_0,y_0)$. Describe the streamlines.

11. ⭐⭐ For $\mathbf{F}(x,y) = \langle x, y\rangle$, solve the streamline system $\dot x = x$, $\dot y = y$ with start $(x_0,y_0)$. Show that $y/x$ is constant along each streamline, so the flow lines are rays from the origin.

12. ⭐⭐ For $\mathbf{F}(x,y) = \langle -y, x\rangle$, verify that $\mathbf{r}(t) = \langle \cos t, \sin t\rangle$ is a streamline by checking $\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$ directly.

13. ⭐⭐ For $\mathbf{F}(x,y) = \langle x, -y\rangle$, solve the streamline system and show the flow lines satisfy $xy = \text{const}$ (hyperbolas). This is the saddle field.

14. ⭐⭐ Explain why solving for streamlines is exactly solving a system of differential equations of the kind introduced in Chapter 19. What plays the role of the "initial condition"?

15. ⭐⭐ If $\mathbf{F} = -\nabla L$ for a loss function $L$, the streamlines are gradient-descent paths (the anchor example from Chapter 6). Where do these streamlines stop, and what does that point represent?

Part C — Divergence (§34.4)

16. ⭐⭐ Compute $\nabla\cdot\mathbf{F}$ for $\mathbf{F}(x,y) = \langle 3x, -3y\rangle$. Classify the point $(2,5)$ as source, sink, or neither.

17. ⭐⭐ Compute the divergence of $\mathbf{F}(x,y,z) = \langle x^2, y^2, z^2\rangle$ and evaluate it at $(1,-1,2)$.

18. ⭐⭐ Show that $\mathbf{F}(x,y) = \langle -y, x\rangle$ is solenoidal (divergence-free). Interpret physically.

19. ⭐⭐ Compute $\nabla\cdot\mathbf{F}$ for $\mathbf{F}(x,y) = \langle x^2 y, -2xy\rangle$ at $(1,3)$ and at $(2,0)$. Classify each point.

20. ⭐⭐ Find all points where $\mathbf{F}(x,y) = \langle xy, x - y^2\rangle$ has zero divergence. (Describe the set.)

21. ⭐⭐ An incompressible fluid has $\nabla\cdot\mathbf{u} = 0$ everywhere. If $\mathbf{u}(x,y) = \langle ax, by\rangle$, what relationship must $a$ and $b$ satisfy?

Part D — Curl (§34.5)

22. ⭐⭐ Compute the (scalar) curl $Q_x - P_y$ for $\mathbf{F}(x,y) = \langle -y, x\rangle$. Interpret the sign.

23. ⭐⭐ Compute the curl of the shear flow $\mathbf{F}(x,y) = \langle y, 0\rangle$. Explain why a paddle wheel spins even though all arrows are parallel.

24. ⭐⭐⭐ Compute the full 3D curl $\nabla\times\mathbf{F}$ for $\mathbf{F}(x,y,z) = \langle yz, xz, xy\rangle$. Is the field irrotational?

25. ⭐⭐⭐ Compute $\nabla\times\mathbf{F}$ for $\mathbf{F}(x,y,z) = \langle x^2, xyz, -y^2\rangle$ and evaluate it at $(1,1,1)$.

26. ⭐⭐⭐ For the magnetic-field model $\mathbf{B}(x,y,z) = \dfrac{1}{x^2+y^2}\langle -y, x, 0\rangle$, show that the scalar curl $Q_x - P_y$ is zero everywhere the field is defined (i.e., away from the $z$-axis). (Hint: this is the vortex pattern.)

Part E — Conservative Fields, the Curl Test, and Potentials (§34.6–34.8)

27. ⭐⭐⭐ Use the curl test to decide whether $\mathbf{F}(x,y) = \langle 2xy, x^2\rangle$ is conservative. If it is, find a potential $f$.

28. ⭐⭐⭐ Use the curl test on $\mathbf{F}(x,y) = \langle y^2 + 1, 2xy\rangle$. If conservative, find a potential.

29. ⭐⭐⭐ Decide whether $\mathbf{F}(x,y) = \langle e^x \cos y, -e^x \sin y\rangle$ is conservative. If so, find $f$.

30. ⭐⭐⭐ Show that $\mathbf{F}(x,y) = \langle 3x^2 + 2y, 2x - 3y^2\rangle$ is conservative and find a potential.

31. ⭐⭐⭐ Show $\mathbf{F}(x,y) = \langle -y, x\rangle$ is not conservative using the curl test. Explain geometrically why a circulating field cannot be a gradient.

32. ⭐⭐⭐ For the 3D field $\mathbf{F} = \langle yz, xz, xy\rangle$ (from problem 24), find a potential $f$ such that $\nabla f = \mathbf{F}$.

33. ⭐⭐⭐ The vortex $\mathbf{F} = \dfrac{1}{x^2+y^2}\langle -y, x\rangle$ has zero curl everywhere it is defined, yet is not conservative on the punctured plane. Which hypothesis of the curl test fails, and why? (See the §34.6 Warning; the full obstruction appears in Chapter 35.)

34. ⭐⭐⭐ A conservative field $\mathbf{F} = \nabla f$ does zero net work around any closed loop. State this fact using the Fundamental Theorem for Line Integrals previewed in §34.8, $\int_C \nabla f\cdot d\mathbf{r} = f(B) - f(A)$, and explain why a closed loop gives $0$.

Part F — Applications and Synthesis (§34.7, §34.9, §34.11–34.13) — Advanced

35. ⭐⭐⭐⭐ (Two fields — fluid and force.) Consider two fields in the plane: $$\mathbf{u}(x,y) = \langle x, -y\rangle \quad\text{(a fluid velocity field)}, \qquad \mathbf{F}(x,y) = -\frac{1}{r^3}\langle x, y\rangle \quad\text{(an inverse-square force, } r=\sqrt{x^2+y^2}).$$ (a) Compute $\nabla\cdot\mathbf{u}$ and the scalar curl of $\mathbf{u}$; classify the flow. (b) Compute $\|\mathbf{F}\|$ as a function of $r$ and state which physical law it models. (c) Show that $\mathbf{F} = \nabla\varphi$ for $\varphi = 1/r$, so $\mathbf{F}$ is conservative; relate $\varphi$ to potential energy (§34.7).

36. ⭐⭐⭐⭐ Prove the identity $\nabla\times(\nabla f) = \mathbf{0}$ for any $f$ with continuous second partials, citing Clairaut's theorem (Chapter 29). Then explain in one sentence why this makes "irrotational" a necessary condition for "conservative."

37. ⭐⭐⭐⭐ Prove the identity $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ for any field $\mathbf{F}$ with continuous second partials. Explain how this identity makes $\nabla\cdot\mathbf{B} = 0$ in Maxwell's equations (§34.12) automatic once $\mathbf{B}$ is written as a curl $\mathbf{B} = \nabla\times\mathbf{A}$.

38. ⭐⭐⭐⭐ A field is harmonic if it is both solenoidal ($\nabla\cdot\mathbf{F}=0$) and irrotational ($\nabla\times\mathbf{F}=\mathbf{0}$). For a conservative harmonic field $\mathbf{F}=\nabla f$, show that the potential satisfies Laplace's equation $\Delta f = f_{xx} + f_{yy} + f_{zz} = 0$. Verify that $f(x,y) = x^2 - y^2$ is harmonic and compute its field $\mathbf{F} = \nabla f$.

Computational Corner (optional, uses Python)

C1. Using the quiver/streamplot template from §34.10, plot the saddle field $\mathbf{F} = \langle x, -y\rangle$. Confirm visually that the streamlines are hyperbolas $xy = \text{const}$ (problem 13).

C2. Using the sympy divergence/curl helpers from §34.10, verify your hand answers to problems 24 and 25. Then test $\nabla\times(\nabla f) = \mathbf{0}$ on a symbolic $f = \texttt{sp.Function('f')(x,y,z)}$ and confirm it returns [0, 0, 0].