| Week | Date | Contents |
|---|---|---|
| 1 | 0903 |
Labour Day |
| 0905 | outline and overview, parametrized curves (by Prof Kim) note 01 |
|
| 0907 | §1.1 derivative of curves, geometric and mechanical meanings,
arclength (by Prof Kim) note 02 |
|
| 2 | 0910 | unit tangent vector, Ex 6 on arclength §1.2 Reparametrization, Ex 1-3 §1.3 curvature, circle, radius and centre of curvature note 03 |
| 0912 | Formula dT/ds = kappa N, Ex 1, 2, formulas for curvature note 04, Ex 2 figures: 1, 2 |
|
| 0914 | formulas for curvature: proof and special cases, Ex 2, 3 §1.4 curves in 3D, binormal vector, Ex 1, osculating plane, Ex 2 note 05 |
|
| 3 | 0917 | The moving frame {T, N, B}, The Frenet-Serret formulas, The
fundamental theorem of space curves, Ex 3 §1.5 summary of formulas §1.6 integrating along a curve note 06 |
| 0919 | Ex 1, integration of vector functions, Ex 2, Ex 3 (we skip §1.7-§1.12) §2.1 vector fields, Ex 1a note 07 |
|
| 0921 | Ex 1b-1d, Ex 2-4. note 08 |
|
| 4 | 0924 | Direction field §2.2 Flow lines Ex 1, Ex 2 (the rest of §2.2 is skipped) §2.3 Conservative vector fields, conservation of energy, Ex 1. note 09 |
| 0926 | Ex 2, flow lines of a conservative vector fields are orthogonal to its
equipotential surfaces, Ex 1, 2 again, necessary condition for a vector field to
be conservative, Ex 3 note 10 |
|
| 0928 |
curl of a vector field, a conservative vector field has zero curl, Ex 4, Ex
3a again, Ex 5: a non-conservative vector
field with zero curl
note 11 |
|
| 5 | 1001 |
Remark on Ex 5 §2.4 Line integral of vector fields, definition, motivation, and remarks, Ex 1 - Ex 3 note 12 |
| 1003 |
Midterm exam 1 | |
| 1005 |
Ex 3b: 3 solutions. §2.4.1 Ex 4, Theorem 1: line integral of a conservative vector field equals the potential difference, hence is path independent note 13 |
|
| 6 | 1008 |
Thanksgiving Day |
| 1010 |
Ex 5. Theorem 2 on equivalent statements of conservative vector
fields. Ex 6. partial proof of Theorem 2. note 14 |
|
| 1012 |
Finish the proof of Theorem 2. Ex 7. Simply connected sets. Theorem
3: A curl-free vector field in a simply connected set is conservative note 15 |
|
| 7 | 1015 |
Summary on available methods to check if a vector field is
conservative. Remark on the general case of Theorem 3. Ex 8. §3.1 parametrized surfaces. Ex 1 - 2. note 16: to be posted later, as I have a visitor this week note 16 |
| 1017 |
Examples 3-9 note 17 |
|
| 1019 |
Finish Ex 9 §3.2 tangent planes, normal vector n = r_u x r_v . Ex 1-3, normal vectors of graphs and level sets note 18 |
|
| 8 | 1022 |
Ex 2 again, Ex 4 §3.3 surface integrals, dS and its vector version dS. Ex 1, 2 for surface area note 19 |
| 1024 |
Remark on area element in polar coordinates, Ex 3, special
case of graphs, Ex 3 again, special case of surfaces of revolution note 20 |
|
| 1026 |
Ex 3 solution 2, Ex 4, Ex 5, surface integral of a scalar function, Ex 6 note 21 |
|
| 9 | 1029 |
Ex 6 finished, Ex 7 and Ex 8. §3.4 flux integrals, physical meaning note 22 |
| 1031 |
physical meaning completed, algebra F · (r_u × r_v) = det
(F, r_u,
r_v), examples 1-4 note 23 |
|
| 1102 |
Ex 5 §3.5 orientation of surfaces, definition, Ex 1 boundary of solid, Ex 2 graph, Ex 3 Mobius strip §4.1 grad, div and curl: notation note 24 |
|
| 10 | 1105 |
§4.1.1 vector identities, Ex 1, 2 §4.1.2 vector potentials, lemmas note 25 |
| 1107 |
Midterm exam 2 |
|
| 1109 |
Ex 3, Theorem: A div-0 vector field in all of R^3 has a vector potential
§4.1.3-5 interpretations of grad, div and curl note 26 |
|
| 11 | 1112 |
Remembrance Day |
| 1114 |
local decomposition of a vector field, Ex 4, 5 §4.2 piecewise smooth surfaces, Ex 1, Divergence theorem: statement and remarks note 27 |
|
| 1116 |
proof of Divergence theorem, special geometry and general geometry.
Ex 2, 3, 4. note 28 |
|
| 12 | 1119 |
Ex 4-8, electric flux note 29 |
| 1121 |
Interpreation of divergence using divergence
theorem,
variations of divergence theorem §4.3 Green's theorem: statement and proof, Ex 1-3. note 30 |
|
| 1123 |
Ex 4-7a note 31 |
|
| 13 |
1126 |
Ex 7bc, remarks, Ex 8,
figure for example 8
§4.4 Stokes' theorem: statement and proof. note 32 |
| 1128 |
Ex 1-3 note 33 |
|
| 1130 |
Ex 4, note 34 conclusion and review |
