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Calculus II, Spring 2025

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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 Vol.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 Vol.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
• (week 12)
  • divergence theorem. reference: 12.19
  • divergence free, but not curl. reference: 12.17
  • application of divergence theorem to change the Laplace in spherical coordinate. reference: Courant & John Vol.II section 5.9.e
• (week 13)
  • elementary probability on a finite sample space. reference: ch.13
  • Stirling formula. reference: Courant & John Vol.I Appendix in ch.6
  • elementary probability on a countable sample space. reference: 13.21
  • uncountable sample spaces, points with positive probability. reference: 14.1, 14.2
  • random variables, distribution functions. reference: 14.3, 14.5, 14.6

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.
• Homework 11 (due 5/15) [12.21] 1, 2, 7, 9, 11.
• Homework 12 (due 5/22) [13.14] 8, 12; [13.18] 12; [14.4] 6; [14.8] 4.