Chapter 34 — Quiz
10 questions covering vector fields, streamlines, divergence, curl, and conservative fields. Try each before opening the answer.
1. Which of the following best distinguishes a vector field from a vector-valued function (Chapter 28)?
(a) A vector field takes a parameter $t$ as input; a vector-valued function takes a point. (b) A vector field assigns a vector to every point of space; a vector-valued function assigns one vector to each value of a parameter $t$. (c) They are the same object written two ways. (d) A vector field outputs a scalar; a vector-valued function outputs a vector.
Answer
**(b).** A vector field $\mathbf{F}(x,y)$ or $\mathbf{F}(x,y,z)$ has a point as input and a vector at *every* point (a carpet of arrows). A vector-valued function $\mathbf{r}(t)$ has a single parameter as input and traces one curve. (§34.1)2. Evaluate the rotational field $\mathbf{F}(x,y) = \langle -y, x\rangle$ at the point $(0,2)$.
Answer
$\mathbf{F}(0,2) = \langle -2, 0\rangle$ — an arrow of length $2$ pointing left. The field circulates counterclockwise about the origin. (§34.2)3. A streamline (flow line) of a vector field $\mathbf{F}$ is a curve $\mathbf{r}(t)$ satisfying which equation?
Answer
$\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$ — the particle's velocity equals the field vector at its current position. Solving this system of ODEs (Chapter 19) produces the streamlines. (§34.3)4. Compute the divergence of $\mathbf{F}(x,y,z) = \langle x, y, z\rangle$.
Answer
$\nabla\cdot\mathbf{F} = \partial_x(x) + \partial_y(y) + \partial_z(z) = 1 + 1 + 1 = 3$. Positive everywhere: every point is a source (radial outflow). (§34.4)5. Compute the divergence of $\mathbf{F}(x,y) = \langle x^2 y,\ -2xy\rangle$ at the point $(1,3)$.
Answer
$P_x = \partial_x(x^2 y) = 2xy$ and $Q_y = \partial_y(-2xy) = -2x$, so $\nabla\cdot\mathbf{F} = 2xy - 2x = 2x(y-1)$. At $(1,3)$: $2(1)(2) = 4 > 0$ — a source. (§34.4)6. Compute the scalar curl $Q_x - P_y$ of the shear flow $\mathbf{F}(x,y) = \langle y, 0\rangle$, and explain the surprise.
Answer
$Q_x = \partial_x(0) = 0$ and $P_y = \partial_y(y) = 1$, so the scalar curl is $0 - 1 = -1$ (i.e., $\nabla\times\mathbf{F} = \langle 0,0,-1\rangle$). Surprise: the arrows are all horizontal and parallel, yet a paddle wheel spins clockwise — the top is pushed harder than the bottom. **Shear is rotation in disguise.** (§34.5)7. Every conservative field is irrotational. State the identity that proves this and name the theorem behind it.
Answer
$\nabla\times(\nabla f) = \mathbf{0}$: the curl of a gradient is always zero. It follows from **Clairaut's theorem** (Chapter 29), equality of mixed partials ($f_{xy} = f_{yx}$). Hence $\nabla\times\mathbf{F}\ne\mathbf{0}$ is an instant proof that $\mathbf{F}$ is *not* conservative. (§34.6)8. Is $\mathbf{F}(x,y) = \langle 2xy,\ x^2\rangle$ conservative? If so, give a potential.
Answer
Curl test: $Q_x = \partial_x(x^2) = 2x$ and $P_y = \partial_y(2xy) = 2x$ — equal, so conservative. Integrate $f_x = 2xy$ in $x$: $f = x^2 y + g(y)$. Then $f_y = x^2 + g'(y) = x^2$, so $g'(y)=0$. Potential: $f(x,y) = x^2 y + C$. (§34.6)9. 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?
Answer
The **simply-connected** hypothesis. The field's domain (the plane minus the origin) has a hole, so a loop around the origin cannot be shrunk to a point. Curl-free guarantees a potential only on a simply connected domain. (Chapter 35 shows the loop integral is $2\pi\ne 0$, proving no potential exists.) (§34.6)10. In Maxwell's equations, $\nabla\cdot\mathbf{B} = 0$. Which vector-calculus identity makes this automatic once $\mathbf{B}$ is written as $\mathbf{B} = \nabla\times\mathbf{A}$, and what is its physical meaning?
Answer
The identity $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ — the divergence of a curl is zero. Physically it says $\mathbf{B}$ has no sources or sinks: there are **no magnetic monopoles**. The deepest meaning of divergence and curl arrives with the Divergence and Stokes theorems in **Chapter 37**. (§34.9, §34.12)Scoring Guide
| Score | Interpretation |
|---|---|
| 9–10 | Excellent — you command vector fields, the two operators, and the conservative test. Move on to line integrals (Ch. 35). |
| 7–8 | Solid. Re-skim the section(s) tied to any miss, especially curl vs. divergence interpretation. |
| 5–6 | Partial. Re-read §34.4–34.6 and redo the worked computations by hand. |
| 0–4 | Revisit the chapter from §34.1. Focus on the field-vs-function distinction and the divergence/curl formulas before retrying. |