Homework and Syllabus
Differential Geometry of Curves and Surfaces
MATH 4230/5230 Fall 2026
⚠ >>WARNING: PAGE UNDER CONSTRUCTION -- SUBJECT TO CHANGE<< ⚠
Read the suggested material before class. After class, attempt the exercises listed for that day. Submit work through Canvas. Clear, logically organized exposition is required for all work in this class.
You may find the Math Academic Resource Center useful as a place to discuss course material and receive additional help.
An asterix * indicates that a homework assignment has not been finalized.
No classHomework due / reviewExam/general reviewMidterm or final
| Date | Topics | Reading | Homework / review material |
|---|---|---|---|
| L1 Friday, August 21 |
Parametrized curves: regularity, speed, and arc length | Tapp §1.1, pp. 2–8 | HW 1(a)Tapp Exercises: 1.5, 1.6, 1.7, 1.8, 1.9, 1.10* |
| L2 Monday, August 24 |
Inner products, projections, and orthonormal bases | Tapp §1.2, pp. 9–15 | HW 1(b)Tapp Exercises: 1.15, 1.16, 1.17, 1.18, 1.19, 1.20* |
| L3 Wednesday, August 26 |
Acceleration and reparametrization | Tapp §§1.3–1.4, pp. 16–23 | HW 1(c)Tapp Exercises: 1.26, 1.27, 1.28, 1.29, 1.31, 1.33* |
| L4 Friday, August 28 |
Curvature, unit tangent and normal vectors, and osculating circles | Tapp §1.5, pp. 24–31 | HW 1 Due HW 2(a)Tapp Exercises: 1.36, 1.37, 1.38, 1.40, 1.41, 1.43* |
| L5 Monday, August 31 |
Plane curves: signed curvature, angle functions, and rotation index | Tapp §1.6, pp. 32–40 | HW 2(b)Tapp Exercises: 1.44, 1.45, 1.46, 1.47, 1.48, 1.49* |
| L6 Wednesday, September 2 |
Space curves: Frenet frame, torsion, and Frenet equations | Tapp §1.7, pp. 41–47 | HW 2(c)Tapp Exercises: 1.62, 1.63, 1.64, 1.65, 1.66, 1.68* |
| L7 Friday, September 4 |
Rigid motions, fundamental theorems, and curvature formulas | Tapp §§1.8–1.9, pp. 48–60 | HW 2 Due HW 3(a)Tapp Exercises: 1.70, 1.71, 1.75, 1.76, 1.78, 1.83* |
| Monday, September 7 | Labor Day | No class; university closed. | No exercises assigned. |
| L8 Wednesday, September 9 |
Hopf’s Umlaufsatz and the Jordan curve theorem | Tapp §2.1, pp. 61–67 | HW 3(b)Tapp Exercises: 1.52, 1.55, 1.58, 2.1, 2.2, 2.3* |
| L9 Friday, September 11 |
Piecewise-regular curves and the generalized Umlaufsatz | Tapp §2.1, pp. 68–71 | HW 3 Due HW 4(a)Tapp Exercises: 1.56, 1.57, 1.59, 1.60, 1.69, 1.82* |
| L10 Monday, September 14 |
Green’s theorem I: vector fields, line integrals, and conservative fields | Tapp §2.4, pp. 81–89 | HW 4(b)Tapp Exercises: 2.17, 2.18, 2.19, 2.20, 2.21, 2.22* |
| L11 Wednesday, September 16 |
Green’s theorem II: circulation, divergence, and flux | Tapp §2.4, pp. 90–96 | HW 4(c)Tapp Exercises: 2.21, 2.23, 2.24, 2.25, 2.26, 2.27* |
| L12 Friday, September 18 |
Review: homework review | Review Tapp Chapters 1–2 | HW 4 Due Six focus problems from the assigned homework: 1.31, 1.38, 1.48, 1.65, 2.18, 2.26. |
| L13 Monday, September 21 |
Review: general review for Midterm I | Review all nonoptional material in Tapp Chapters 1–2 | Six cumulative review problems: 1.10, 1.20, 1.33, 1.52, 1.64, 2.24. |
| L14 Wednesday, September 23 |
Review: Midterm I practice exam | Practice exam and solutions will be posted on Canvas. | Complete the six-problem practice exam before class; bring questions. |
| L15 Friday, September 25 |
MIDTERM I | Tentative scope: nonoptional material in Tapp Chapters 1–2 | No new exercises assigned. |
| L16 Monday, September 28 |
Derivatives of maps from Euclidean space to Euclidean space | Tapp §3.1, pp. 113–124 | HW 5(a)Tapp Exercises: 3.1, 3.2, 3.3, 3.4, 3.5, 3.9* |
| L17 Wednesday, September 30 |
Review Midterm I | Review the returned exam, common errors, and solution strategies. | No new exercises assigned; bring your exam and corrected solutions. |
| L18 Friday, October 2 |
Regular surfaces: patches, level sets, ruled surfaces, and examples | Tapp §3.2, pp. 125–140 | HW 5 Due HW 6(a)Tapp Exercises: 3.13, 3.15, 3.18, 3.20, 3.23, 3.29* |
| L19 Monday, October 5 |
Tangent planes and derivatives of maps between surfaces | Tapp §3.3, pp. 141–146 | HW 6(b)Tapp Exercises: 3.35, 3.36, 3.38, 3.40, 3.43, 3.46* |
| L20 Wednesday, October 7 |
Area distortion, orientation, and orientable surfaces | Tapp §§3.4–3.5, pp. 147–159 | HW 6(c)Tapp Exercises: 3.51, 3.54, 3.57, 3.59, 3.63, 3.72* |
| L21 Friday, October 9 |
Surface area, isometries, and the first fundamental form | Tapp §§3.6–3.7, pp. 160–169 | HW 6 Due HW 7(a)Tapp Exercises: 3.74, 3.76, 3.79, 3.80, 3.83, 3.85* |
| L22 Monday, October 12 |
The first fundamental form in local coordinates | Tapp §3.9, pp. 182–187 | HW 7(b)Tapp Exercises: 3.100, 3.101, 3.103, 3.105, 3.107, 3.108* |
| L23 Wednesday, October 14 |
The Gauss map and the Weingarten map | Tapp §4.1, pp. 195–200 | HW 7(c)Tapp Exercises: 4.1, 4.2, 4.3, 4.4, 3.102, 3.104* |
| L24 Friday, October 16 |
Self-adjoint maps, the spectral theorem, and principal directions | Tapp §4.2, pp. 201–205 | HW 7 Due HW 8(a)Tapp Exercises: 4.5, 4.6, 4.7, 4.8, 3.106, 3.109* |
| L25 Monday, October 19 |
Normal curvature, principal curvatures, Gaussian curvature, and mean curvature | Tapp §4.3, pp. 206–212 | HW 8(b)Tapp Exercises: 4.9, 4.10, 4.12, 4.14, 4.16, 4.18* |
| L26 Wednesday, October 21 |
Geometric characterizations of Gaussian curvature | Tapp §4.4, pp. 213–216 | HW 8(c)Tapp Exercises: 4.19, 4.20, 4.21, 4.22, 4.23, 4.24* |
| L27 Friday, October 23 |
Review: homework review | Review Tapp Chapters 3–4 through §4.4 | HW 8 Due Six focus problems from the assigned homework: 3.9, 3.23, 3.43, 3.72, 3.105, 4.18. |
| L28 Monday, October 26 |
The second fundamental form in local coordinates | Tapp §4.5, pp. 217–226 | HW 9(a)Tapp Exercises: 4.25, 4.28, 4.30, 4.31, 4.33, 4.40* |
| L29 Wednesday, October 28 |
Review: general review for Midterm II | Review all nonoptional material in Tapp Chapters 3–4 | Six cumulative review problems: 3.5, 3.29, 3.46, 3.79, 4.8, 4.24. |
| L30 Friday, October 30 |
Review: Midterm II practice exam | Practice exam and solutions will be posted on Canvas. | HW 9 Due Complete the six-problem practice exam before class; bring questions. |
| L31 Monday, November 2 |
MIDTERM II | Tentative scope: nonoptional material in Tapp Chapters 3–4 | No new exercises assigned. |
| L32 Wednesday, November 4 |
Geodesics: definitions, examples, and Clairaut’s theorem | Tapp §5.1, pp. 247–256 | HW 10(a)Tapp Exercises: 5.1, 5.2, 5.4, 5.5, 5.7, 5.9* |
| L33 Friday, November 6 |
The exponential map, Gauss’s lemma, and normal coordinates | Tapp §5.2, pp. 257–267 | HW 10 Due HW 11(a)Tapp Exercises: 5.21, 5.22, 5.23, 5.24, 5.27, 5.32* |
| L34 Monday, November 9 |
Gauss’s Theorema Egregium, intrinsic curvature, and complete surfaces | Tapp §§5.3–5.4, pp. 268–279 | HW 11(b)Tapp Exercises: 5.33, 5.35, 5.39, 5.41, 5.43, 5.47* |
| L35 Wednesday, November 11 |
Review Midterm II | Review the returned exam, common errors, and solution strategies. | No new exercises assigned; bring your exam and corrected solutions. |
| L36 Friday, November 13 |
Parallel transport and the covariant derivative | Tapp §5.5, pp. 280–288 | HW 11 Due HW 12(a)Tapp Exercises: 5.48, 5.49, 5.50, 5.52, 5.54, 5.55* |
| L37 Monday, November 16 |
Geodesics in local coordinates and curvature as infinitesimal holonomy | Tapp §§5.6–5.7, pp. 289–302 | HW 12(b)Tapp Exercises: 5.56, 5.58, 5.59, 5.64, 5.69, 5.71* |
| L38 Wednesday, November 18 |
The local and global Gauss–Bonnet theorems | Tapp §§6.1–6.2, pp. 320–333 | HW 12(c)Tapp Exercises: 6.1, 6.2, 6.3, 6.4, 6.7, 6.9* |
| L39 Friday, November 20 |
Compact surfaces, Euler characteristic, and other global theorems | Tapp §§6.3–6.4, pp. 334–344 | HW 12 Due HW 13(a)Tapp Exercises: 6.10, 6.12, 6.15, 6.16, 6.18, 6.22* |
| Monday, November 23 | Fall Break | No class. | No exercises assigned. |
| Wednesday, November 25 | Fall Break | No class. | No exercises assigned. |
| Friday, November 27 | Fall Break / Thanksgiving holiday | No class; university closed. | No exercises assigned. |
| L40 Monday, November 30 |
Review: homework review | Review Tapp Chapters 5–6 | Six focus problems from the assigned homework: 5.9, 5.23, 5.39, 5.50, 5.69, 6.9. |
| L41 Wednesday, December 2 |
Review: general review for the final examination | Cumulative review of all nonoptional sections in Tapp Chapters 1–6 | Six cumulative review problems: 1.38, 2.24, 3.46, 4.31, 5.47, 6.15. |
| L42 Friday, December 4 |
Review: final practice exam and course synthesis | Six-problem cumulative practice final and solutions will be posted on Canvas. | HW 13 Due Complete the six-problem practice final before class; bring questions. |
| Tuesday, December 8 | FINAL EXAMINATION | 1:30–4:00 PM Room listed in Buff Portal |
No new exercises assigned. |
This schedule is subject to adjustment.