C A L C U L U S
I I ,
S P R I N G 2 0 2 5
Course Information
- NTU COOL
- Lectures: Tuesday and Thursday, 13:20 ~ 15:10 at Freshman Classroom Building 304
- Office hours: Wednesday 14:00 ~ 15:00, at Astro-Math 458
- Grading scheme:
- Midterm 30% (on April 10)
- Final 35% (on June 5)
- Quiz 10% (on March 13 and May 15)
- Homework 25%
You have two jokers: the lowest two grades will be discarded.
- Textbook: Tom Apostol, Calculus vol.2 - Multi-Variable Calculus and Linear Algebra with Application to Differential Equations and Probability.
- Problem session: Thursday 17:30 ~ 18:20
- Teaching Assistants:
- 連焌凱: problem session at Freshman Classroom Building 103. Office hour: Wednesday 13:20 ~ 14:20 at Astro-Math 446.
- 郭立生: problem session at Freshman Classroom Building 304. Office hour: Wednesday 15:30 ~ 16:30 at Astro-Math 405.
- 黃篆: problem session at Freshman Classroom Building 203. Office hour: Friday 17:30 ~ 18:30 at Astro-Math 449.
Lecture summaries and references
- (week 1)
- basic point-set topology of Euclidean space. reference: 8.2
- limits and continuity. reference: 8.4
- partial derivative and directional derivative. reference: 8.6 ~ 8.8
- derivative and continuity. reference: 8.10, and Courant&John V.II p.34 ~ 35
- commuting second order derivatives. reference: 8.23
- (week 2)
- differentiability in multi-variables. reference: 8.11 ~ 8.13, 8.19
- chain rule. reference: 8.20, 8.21
- tangent plane of a graph. reference: 8.16
- Taylor theorem in multi-variables. reference: 9.10
- local extremal and 2nd derivative test. reference: 9.9, 9.11, 9.12
- (week 3)
- eigenvalues and eigenvectors of 2x2 symmetric matrices.
- implicit differentiation. reference: 9.6, 9.7
- implicit function theorem. reference: note on NTU COOL
- Lagrange multiplier. reference: 9.14
- (week 4)
- Lagrange multiplier (continued). reference: 9.14
- uniform continuity and consequences in multi-variables. reference: 9.16, 9.17
- arc-length of a curve. reference: (Vol. 1) 14.10, 14,11
- vector line integral. reference: 10.2 ~ 10.4
- fundamental theorem of vector line integral. reference: 10.14
- (week 5)
- necessary and sufficient condition for a gradient vector field. reference: 10.5 ~ 10.7
- construction of the potential function. reference: 10.21
- double integral. repeated integral. reference: 11.2 ~ 11.8
- Jordan measure (content). reference: note on NTU COOL
- (week 6)
- Jordan measure (content) and Riemann integrability for multiple integrals. reference: note on NTU COOL
- a theorem about integrability. reference: 11.11
- Fubini theorem: continuous function over rectangle. reference: 11.10
- double integral over Type I and Type II regions. reference: 11.12, 11.14
- 2D Green's theorem. reference: 11.19 ~ 11.21
- (week 7)
- operator norm of linear transformations. reference: baby Rudin ch.9
- contraction mapping principle. reference: baby Rudin ch.9
- inverse and implicit function theorems. reference: baby Rudin ch.9
- MIDTERM
- (week 8)
- change of variable formula in 2D. reference: 11.26, 11.29
- polar coordinate. reference: 11.27
- (week 9)
- cylindrical and spherical coordinate. reference: 11.33
- volume of the n-dimensional ball. reference: 11.33
- improper integral in multi-variables. reference: Courant&John V.II section 4.7
- surface area: Schwarz lantern. reference: see wikipedia
- (week 10)
- surface and parametrization. reference: 12.1
- tangent plane. reference: 12.2, 12.3
- surface area. reference: 12.5
- surface area: independent of parametrization. reference: 12.8
- (scalar) surface integral. reference: 12.7
- orientation of surface. reference: 12.9
- (week 11)
- flux integral. reference: 12.9
- curl of a vector field. reference: 12.12
- Stokes theorem. reference: 12.11
- divergence of a vector field. reference: 12.12, 12.14
- reconstructing a vector field from its curl. reference: 12.16
Homework
- Homework 01 (due 2/27) [8.3] 2 (m), 5 (c), (d), (e); [8.5] 1 (j), 3, 4; [8.8] 8, 14.
- Homework 02 (due 3/06) [8.14] 2 (b), 8 (a) to (d); [8.17] 4 (a), (b); [8.22] 5, 14; [8.24] 7 (a), (b).
- Homework 03 (due 3/13) [9.8] 4; [9.13] 14; [9.15] 6.
- Homework 04 (due 3/20) [9.15] 8, 11; [10.5] 11, 12 (a), (b); [Vol.1 14.13] 13 (a), (b).
- Homework 05 (due 3/27) [10.9] 3; [10.13] 5 (b), 7 (a), (b); [10.18] 3, 11, 15.
- Homework 06 (due 4/08) [11.9] 14; [11.15] 3, 17, 18, 19, 22.
- Homework 07 (due 4/17) [11.22] 1 (a), 2, 6 (a), (b).
- Homework 08 (due 4/24) [11.28] 14, 16 (a), (b), (c), 19; [11.34] 4, 9, 15.
- Homework 09 (due 5/01) [12.4] 3, 6; [12.6] 2, 4, 9, 11.
- Homework 10 (due 5/08) [12.10] 7, 14; [12.13] 4, 6, 12; [12.17] 2.
Last modified: May 4, 2025.
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