Chapter 35 — Quiz

Q: The scalar line integral is computed with which element? B) C) D) just

B) , the arc-length element. Forgetting the factor is the single most common error in the chapter: must return the length of . Section 35.2.

Q: A vector line integral uses which integrand in the parameter ? B) C) D)

B) — a dot product, giving the tangential component of the force times speed. This is work in physics. Section 35.3.

Q: Reversing the orientation of a curve affects the two flavors of line integral how? Both flip sign B) Neither changes C) Scalar unchanged; vector flips sign D) Vector unchanged; scalar flips sign

C). A scalar line integral measures an arc-length "curtain" and is blind to direction; a vector line integral negates , hence flips sign: . Section 35.4.

Q: For a conservative field and a curve from to , the value of is: always B) C) dependent on the path D) undefined

B) — the Fundamental Theorem for Line Integrals. The integral depends only on the endpoints, not the route. This is the original FTC (Chapter 14) generalized to curves. Section 35.5.

Q: For a conservative field , the circulation around any closed loop equals: B) the enclosed area C) the curl at the center D) it depends on the loop

A) . On a closed loop , so . This closed-loop corollary is the mathematical face of conservation of energy. Section 35.5.

Q: On a simply-connected domain, which set of conditions is equivalent to being conservative? curl-free B) path-independent C) for every closed D) all of the above

D) all of the above. Conservative curl-free path-independent zero circulation on every loop — but only on a simply-connected domain. Section 35.6.

Q: Green's theorem states . Fill the blank. B) C) D)

C) , the 2D curl (the -component of ). The boundary's total circulation equals the interior's total microscopic curl. Section 35.7.

Q: The area of a region can be recovered from its boundary by: B) C) D)

B) . Choosing , gives , so the double integral becomes the area. This is the principle of the planimeter and the continuous shoelace formula. Section 35.7.

Q: The field is curl-free yet has around the unit circle. Why is this not a contradiction? The curl test is wrong B) Its domain (punctured plane) is not simply connected C) The circle is not closed D) It is actually conservative

B). Curl-free implies conservative only on simply-connected domains. The domain has a hole, and the loop encircling the origin cannot shrink to a point. Section 35.6, Math Major Sidebar.

Q: A radial force field does how much work moving a particle once around a full circle centered at the origin? B) depends on radius C) D)

C) . The radial force is everywhere perpendicular to the circular (tangential) motion, so at every instant. (It is also conservative, , so any closed loop gives .) Sections 35.3, 35.5.

DataField.Dev

Chapter 35 — Quiz

10 questions covering scalar and vector line integrals, the Fundamental Theorem for Line Integrals, path independence, and Green's theorem. Try each before opening the answer; every answer cites the section to review.

1. The scalar line integral $\int_C f\,ds$ is computed with which element?

A) $\mathbf{r}'(t)\,dt$ B) $|\mathbf{r}'(t)|\,dt$ C) $\mathbf{F}\cdot\mathbf{r}'\,dt$ D) just $dt$

Answer

**B) $|\mathbf{r}'(t)|\,dt = ds$**, the arc-length element. Forgetting the $|\mathbf{r}'|$ factor is the single most common error in the chapter: $\int_C 1\,ds$ must return the *length* of $C$. *Section 35.2.*

2. A vector line integral $\int_C\mathbf{F}\cdot d\mathbf{r}$ uses which integrand in the parameter $t$?

A) $f\,|\mathbf{r}'|$ B) $\mathbf{F}\cdot\mathbf{r}'$ C) $\mathbf{F}\times\mathbf{r}'$ D) $|\mathbf{F}|\,|\mathbf{r}'|$

Answer

**B) $\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)$** — a dot product, giving the tangential component of the force times speed. This is *work* in physics. *Section 35.3.*

3. Reversing the orientation of a curve $C$ affects the two flavors of line integral how?

A) Both flip sign B) Neither changes C) Scalar unchanged; vector flips sign D) Vector unchanged; scalar flips sign

Answer

**C).** A scalar line integral measures an arc-length "curtain" and is blind to direction; a vector line integral negates $\mathbf{r}'$, hence flips sign: $\int_{-C}\mathbf{F}\cdot d\mathbf{r}=-\int_C\mathbf{F}\cdot d\mathbf{r}$. *Section 35.4.*

4. For a conservative field $\mathbf{F}=\nabla f$ and a curve from $A$ to $B$, the value of $\int_C\mathbf{F}\cdot d\mathbf{r}$ is:

A) always $0$ B) $f(B)-f(A)$ C) dependent on the path D) undefined

Answer

**B) $f(B)-f(A)$** — the Fundamental Theorem for Line Integrals. The integral depends only on the endpoints, not the route. This is the original FTC (Chapter 14) generalized to curves. *Section 35.5.*

5. For a conservative field $\mathbf{F}$, the circulation $\oint_C\mathbf{F}\cdot d\mathbf{r}$ around any closed loop equals:

A) $0$ B) the enclosed area C) the curl at the center D) it depends on the loop

Answer

**A) $0$.** On a closed loop $A=B$, so $f(B)-f(A)=0$. This closed-loop corollary is the mathematical face of conservation of energy. *Section 35.5.*

6. On a simply-connected domain, which set of conditions is equivalent to $\mathbf{F}$ being conservative?

A) curl-free B) path-independent C) $\oint_C\mathbf{F}\cdot d\mathbf{r}=0$ for every closed $C$ D) all of the above

Answer

**D) all of the above.** Conservative $\iff$ curl-free $\iff$ path-independent $\iff$ zero circulation on every loop — but only on a simply-connected domain. *Section 35.6.*

7. Green's theorem states $\oint_C P\,dx+Q\,dy=\iint_D(\underline{\quad})\,dA$. Fill the blank.

A) $P+Q$ B) $P_x+Q_y$ C) $Q_x-P_y$ D) $P_y-Q_x$

Answer

**C) $Q_x-P_y$**, the 2D curl (the $z$-component of $\nabla\times\mathbf{F}$). The boundary's total circulation equals the interior's total microscopic curl. *Section 35.7.*

8. The area of a region $D$ can be recovered from its boundary by:

A) $\oint_C \mathbf{F}\cdot\mathbf{n}\,ds$ B) $\tfrac12\oint_C(x\,dy-y\,dx)$ C) $\oint_C f\,ds$ D) $\iint_D(P_y-Q_x)\,dA$

Answer

**B) $\tfrac12\oint_C(x\,dy-y\,dx)$.** Choosing $P=-y/2$, $Q=x/2$ gives $Q_x-P_y=1$, so the double integral becomes the area. This is the principle of the planimeter and the continuous shoelace formula. *Section 35.7.*

9. The field $\mathbf{F}=\langle -y,x\rangle/(x^2+y^2)$ is curl-free yet has $\oint_C\mathbf{F}\cdot d\mathbf{r}=2\pi$ around the unit circle. Why is this not a contradiction?

A) The curl test is wrong B) Its domain (punctured plane) is not simply connected C) The circle is not closed D) It is actually conservative

Answer

**B).** Curl-free implies conservative *only* on simply-connected domains. The domain $\mathbb{R}^2\setminus\{0\}$ has a hole, and the loop encircling the origin cannot shrink to a point. *Section 35.6, Math Major Sidebar.*

10. A radial force field $\mathbf{F}=\langle x,y\rangle$ does how much work moving a particle once around a full circle centered at the origin?

A) $2\pi$ B) depends on radius C) $0$ D) $\pi r^2$

Answer

**C) $0$.** The radial force is everywhere perpendicular to the circular (tangential) motion, so $\mathbf{F}\cdot\mathbf{r}'=0$ at every instant. (It is also conservative, $\nabla(\tfrac12(x^2+y^2))$, so any closed loop gives $0$.) *Sections 35.3, 35.5.*

Scoring

9–10 correct — Excellent. You command both flavors of line integral and both fundamental theorems of the chapter.
7–8 correct — Solid. Revisit any missed item's cited section.
Below 7 — Review §35.2–35.3 (the two integral types and their elements), §35.5 (the Fundamental Theorem for Line Integrals), and §35.7 (Green's theorem). Re-work the chapter's ten worked examples by hand.