Chapter 38 — Quiz

10 questions on differential forms and the generalized Stokes' theorem. Section references point to the chapter text.

1. FTC, the line-integral FTC, Green's, Stokes', and Divergence are all special cases of: - A) Limits - B) The original FTC alone - C) The generalized Stokes' theorem $\int_{\partial M}\omega = \int_M d\omega$ - D) Both B and C

Answer**D) Both B and C.** Each is a higher-dimensional version of FTC, and all are unified by the generalized Stokes' theorem $\int_{\partial M}\omega = \int_M d\omega$. *Section 38.1, Section 38.4.*

2. True or False: the master theorem reads $\int_{\partial M}\omega = \int_M d\omega$. - A) True - B) False

Answer**A) True.** The integral of $\omega$ over the boundary equals the integral of $d\omega$ over the region. *Section 38.4.*

3. A differential $1$-form is the natural thing to integrate over: - A) Points - B) Curves - C) Surfaces - D) Solids

Answer**B) Curves.** A $k$-form integrates over a $k$-dimensional region; a $1$-form gives a line integral. *Section 38.2.*

4. A differential $2$-form is the natural thing to integrate over: - A) Points - B) Curves - C) Surfaces - D) Solids

Answer**C) Surfaces.** A $2$-form is the integrand of a flux integral. *Section 38.2.*

5. The exterior derivative $d$ takes a $k$-form to: - A) A $(k-1)$-form - B) A $k$-form - C) A $(k+1)$-form - D) A $0$-form

Answer**C) A $(k+1)$-form.** Raising the degree by one is exactly the bump needed to go from the boundary to the region. *Section 38.3.*

6. Applying $d$ to a $1$-form built from a vector field $\mathbf{F}$ reproduces which classical operator? - A) Gradient - B) Curl - C) Divergence - D) Laplacian

Answer**B) Curl.** $d$ on a $0$-form gives the gradient, on a $1$-form gives the curl, on a $2$-form gives the divergence. *Section 38.3.*

7. The identity $d^2 = 0$ says: - A) $d$ has no effect - B) Applying $d$ twice always gives zero - C) $d$ equals the identity - D) $d$ is undefined

Answer**B) Applying $d$ twice always gives zero.** It encodes both $\nabla\times(\nabla f)=\mathbf{0}$ and $\nabla\cdot(\nabla\times\mathbf{F})=0$. *Section 38.6.*

8. A closed form satisfies: - A) $d\omega \neq 0$ - B) $d\omega = 0$ - C) $\omega = 0$ - D) $\omega = d\eta$

Answer**B) $d\omega = 0$.** (A form with $\omega = d\eta$ is called *exact*.) *Section 38.6.*

9. True or False: every exact form is closed. - A) True - B) False

Answer**A) True.** If $\omega = d\eta$ then $d\omega = d(d\eta) = 0$ by $d^2 = 0$. The converse can fail on a region with a hole. *Section 38.6.*

10. In the language of differential forms, Maxwell's four equations of electromagnetism collapse to how many equations? - A) 4 - B) 2 ($dF = 0$ and $d{\star}F = J$) - C) 1 - D) 8

Answer**B) 2.** $dF = 0$ packages the two source-free laws (and is automatic from $d^2 = 0$ once $F = dA$); $d{\star}F = J$ packages the two laws with sources. *Section 38.8.*

Scoring Guide

  • 9–10 correct: Excellent. You have internalized the synthesis — forms, the exterior derivative, and the master theorem.
  • 7–8 correct: Solid. Revisit form degrees (Section 38.2) and the recovery of the classical theorems (Section 38.5).
  • 5–6 correct: Review the exterior derivative and $d^2 = 0$ (Sections 38.3 and 38.6).
  • Below 5: Re-read the chapter, focusing on the master-theorem table in Section 38.5.