Calculus II, Spring 2022.

A undergraduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang (王偉成).

Office Hours:
T,R: 13:00-14:00 via Google meet or by appointment.

Teaching Assistant:

• B05: 邱志文 (lmp127552@gmail.com).
• B08: 李長紘 (alex60901@gmail.com).

Lecture: Delta building (台達館), classrooms B03, T3T4R3R4.

Recitation: Delta building (台達館), classrooms B05 ( B05 Google meet link) and B08 ( B08 Google meet link) Tuesdays 19:00-21:00.

Grading: 40% quiz(pick n-1 best from n quizzes)+attendance, 20% midterm I, 20% midterm II + 20% final exam. Extra credits for significant contributions.

Textbook: G. B. Thomas, M. D. Weir and J. R. Hass: Thomas's Calculus Early Transcendentals, 13th edition in SI units (華通書局有販售,詳見校務資訊系統課程大綱).

Contents:

• Calculus II:
• Chap 10: Infinite Sequences and Series.
• Chap 14: Partial Derivatives.
• Chap 15: Multiple Integrals.
• Chap 16: Integration in Vector Fields.

Syllabus and study guide:

• Calculus II:
• Syllabus (revised May 30).
• quiz 01 study guide. quiz 01 solutions. (average = 70.04 pts)
  
• quiz 02 study guide. quiz 02 solutions. (average = 72.56 pts)
  
• quiz 03 study guide v02 (updated). quiz 03 solutions. (average = 77.66 pts)
  
• Midterm 01 study guide. Midterm 01 solutions. (average = 49.04 pts)
  
• quiz 04 study guide (updated). quiz 04 solutions. (average = 75.19 pts)
  
• quiz 05 study guide. (average = 74.00 pts)
• quiz 06 study guide. quiz 06 solutions. (average = 90.02 pts)
  
• Midterm 02 study guide. Midterm 02 solutions (v02, add more details on partial credits). (average = 75.41 pts)
  
• Chap 15 study guide.
• Chap 16 study guide (v03: up to Section 16.8, completed).
• Remote quiz 09 solutions. (average = 58.40 pts)
  
• Remote quiz 10 solutions. (average = 61.50 pts)
  
• Remote Final Exam solutions. (average = 52.18 pts)
  
• Semester grade formula.
  

Video Links, lecture notes, homework assignments and exam solutions:

• Week 01: No recitation this week.
  Lecture 01 (20220215): Section 8.8: Definition of improper integrals, type I and type II. Integrals ∫₀¹ x^-p dx and ∫₁^∞ x^-p dx. Test convergence and divergence.
  Lecture 01 note.
  Lecture 02 (20220217): Section 8.8: Direct Comparison Test and Limit Comparison Test. Section 10.1: Definition of limit of sequences, including the limit = +- ∞ cases.
  Lecture 02 note.
  Homework week 01. Homework week 01 solutions.
  
• Week 02: Recitation starts this week (Tbc).
  Lecture 03 (20220222): Chapter 10: The role of sequences and infinite series in Calculus. Section 10.2: Infinite series, definition and example of convergent and divergent series. The n-th term test. Section 10.3: The integral test, Proof and examples. Section 10.4: Direct comparison test.
  Lecture 03 note.
  Lecture 04 (20220224): Section 10.4: Limit comparison test. Examples for both Direct and Limit comparison tests. Section 10.5: Ratio and root test. Proof of ratio test. Examples of both tests.
  Lecture 04 note (New! additional explanation on page 2 and page 5). Homework week 02 (updated, added Section 10.3, problem 57 (optional)).
  Homework week 02 solutions.
  
• Week 03: Quiz 01 (Tuesday 03/01 10:10AM).
  Lecture 05 (20220301): Section 10.6: Alternating series: Leibniz test. Definition of absolute and conditional convergence. Absolute convergence test. Examples. Ratio and root test for general (not necessarily positive a_n) series.
  Lecture 05 note (added page 9).
  Lecture 06 (20220303): Section 10.7: Definition and examples of power series, including geometric series; The convergence Theorem for power series; Finding radius of convergence using ratio and root test; Multiplication and division of power series.
  Lecture 06 note (revised on page 6).
  Homework week 03. Homework week 03 solutions.
  
• Week 04: Quiz 02 (Tuesday 03/08 10:10AM).
  Lecture 07 (20220308): Section 10.7: Term by term differentiation of power series; Examples, counter examples (not power series); Variants of geometric series using term by term differentiation. Term by term integration; Examples of tan^-1(x) and ln(1±x).
  Lecture 07 note (Updated. Added power series representation of ln(1±x) on page 8.
  Lecture 08 (20220310): Section 10.8: Taylor and MacLaurin series. Definition of T_f,a(x). Example of T_f,a(x) ≠ f(x). Examples on how to find T_f,a(x) with f(x)= e^x, cos(x), sin(x). Application in numerical integration using truncated Taylor series and error estimate.
  Lecture 08 note.
  Homework week 04. Homework week 04 solutions.
  
• Week 05: Quiz 03 (Tuesday 03/15 10:10AM).
  Lecture 09 (20220315): Section 10.9: Taylor's Theorem (Taylor's formula)and proof. Corollary on how to check whether T_f,a(x) = f(x). Sufficient condition on estimates of the remainder term in Taylor's formula. How to find the power series representation of f (if it exists) using derivatives of f.
  Lecture 09 note.
  Lecture 10 (20220317): Section 10.10: Binomial series. Application to Taylor series generated by sin^-1(x). find power series expansion of f(x)/g(x) by long division. Finding the sum of a power series by identifying it with the Taylor series of basic functions. Applications to evaluation of intermediate forms and Euler identity of exp(iθ).
  Lecture 10 note.
  Homework week 05. Homework week 05 solutions.
  
• Week 06: Midterm 01 (Thursday 03/24 10:10AM).
  Lecture 11 (20220322): Section 10.10: Remarks on Leibniz's formula for π/4 and ln 2. Section 14.2: Definition of lim_{(x,y)→(x0,y0)} f(x,y). Example of finding δ from a given ε for a simple f(x,y).
  Lecture 11 note.
  Homework week 06. Homework week 06 solutions.
  
• Week 07:
  Lecture 12 (20220329): fig 14.13, fig 14.14, fig 14.15.
  Section 14.3: Several examples of lim_{(x,y)→(x0,y0)} f(x,y) does not exist and two path Theorem; Continuity of f(x,y). Examples of discontinuity at a point. Continuity of composition of continuous functions. Partial derivatives. Implicit partial differentiation. Defining differentiability in terms of existence of a tangent plane.
  Lecture 12 note.
  Lecture 13 (20220331): Section 14.3: Further explanation of tangency of two curves in a plane and two surfaces in space. Showing the existence of a tangent plane leads to the definition of continuity given on the textbook. Sufficient condition of differentiability in terms of continuity of first partial derivatives. Differentiability implies continuity. Continuity of second partial derivatives implies interchanging order of two partial derivatives and a counter example. Section 14.4: Application of differentiability to the Chain rule.
  Lecture 13 note (v02, page 6 revised).
  Homework week 07. Homework week 07 solutions.
  
• Week 08:
  Lecture 14 (20220407): Section 14.4: Implicit partial differentiation revisited. Section 14.5: Definition and examples of directional derivative. Differentiability leads to formula of directional derivative and counter example. Properties of ∇f. Finding tangent and normal lines of a level curve of F(x,y) using the gradient vector ∇F.
  Lecture 14 note.
  Homework week 08 (v02, minor revision). Homework week 08 solutions.
  
• Week 09: Quiz 04 (Tuesday 04/12 10:10AM).
  Lecture 15 (20220412): Section 14.5: Find tangent plane and normal line of a level surface of F(x,y,z). Find tangent line and normal plane of a curve in space. Section 14.6: Relation between Linearization and tangency. Equivalent characterization of differentiability.
  Lecture 15 note.
  Lecture 16 (20220414): Section 14.6: Further explanation on relations among continuity, directional derivative and differentiability (homework 08 problem 2). Section 14.7: Extreme values and saddle points. The second derivative test.
  Lecture 16 note.
  Homework week 09.
• Week 10: Quiz 05 (Tuesday 04/19 10:10AM).
  Lecture 17 (20220419): Section 14.7: The second derivative test continued. Alternative to second derivative test: gradient analysis. Finding global extrema using gradient analysis. Section 14.8: Constrained optimization.
  Lecture 17 note.
  Lecture 18 (20220421): Section 14.8: Constrained optimization continued, various examples. Section 14.9: Derivation of Taylor formula for f(x,y).
  Lecture 18 note.
  Homework week 10. Homework week 10 solutions.
  
• Week 11: Quiz 06 (Tuesday 04/26 10:10AM).
  Lecture 19 (20220426):
  Section 14.10: Partial Derivatives with Constrained variables. Examples. Section 15.1: Partial Derivatives with Constrained variables. Examples. Double integrals defined as limit of Riemann sum. Fubini's Theorem and iterated integrals on rectangular regions. Section 15.2: Fubini's Theorem and iterated integrals on rectangular-like regions.
  Lecture 19 note.
  Lecture 20 (20220428): Section 15.2: Interchanging order of integration for double (iterated) integrals. Section 15.3: Area by double integrals, special case of section 15.2, skipped. Section 15.4: Integration in polar form. Derivation of dA = r dr dθ.
  Lecture 20 note.
  Homework week 11. Homework week 11 solutions.
  
• Week 12: Midterm 02 (Thursday 05/05 10:10AM).
  Lecture 21 (20220503): Section 15.4: Identifying limits of integration in polar form. Examples. Section 15.5: Triple integrals in rectangular coordinates. Identifying limits of integration in iterated triple integrals.
  Lecture 21 note.
  Homework week 12. Homework week 12 solutions.
  
• Week 13: No quiz this week.
  Lecture 22 (20220510): Section 15.5: Identifying limits of integration in iterated triple integrals continued. Section 15.7: Cylindrical coordinates. Identifying limits of integration in iterated triple integrals in cylindrical coordinates.
  Lecture 22 note.
  Lecture 23 (20220512): Section 15.7: Details of spherical coordinates. Derivation of dV = ρ² sin φ dρ dφ dθ. Section 15.8: Derivation of substitutions in multiple integrals. Examples in 2D.
  Lecture 23 note (v2, minor revisions on page 2 and page 5).
  Homework week 13. Homework week 13 solutions.
  
• Week 14: quiz 7 cancelled.
  Lecture 24 (20220517): Section 15.7: Remark on coordinate change (r,z) <---> (ρ,φ). Section 16.1: Definition of line integrals as limit of Riemann sum. Computation of scalar line integrals by means of parametrization of the curve. Showing the result is independent of parametrization. Section 16.2: Related integrals: Work done along the path.
  Lecture 24 note (with remarks on homework 13 in the beginning).
  Lecture 25 (20220519): Section 16.2: Computation of Work done along the path by means of correctly oriented parametrization of the curve. Explanation of simple closed curves. Notation of circulation and flux over a simple closed curve. Section 16.3: Fundamental Theorem of line integrals. Statement and proof. Explanation of connected and simply connected regions. Equivalence of conservative field and existence of potential function. Loop property of conservative fields. Component Test and exact fields. Finding potential functions for exact fields in simply connected regions.
  Lecture 25 note (v02, correction on p09-p10). Update: v03, correction (-x'(t)) and remark on p03. .
  Homework week 14. Homework week 14 solutions.
  
• Week 15: quiz 8 cancelled.
  Lecture 26 (20220524): Section 16.3: Summarize equivalent characterizations of Conservative Fields. Examples of domains which are and are not simply connected. A very important example of an exact vector field which is not conservative on a non-simply connected domain. Section 16.4: Statement of Stokes' Theorem and The Divergence Theorem. Explain the notations used in these Theorems. Reduction of these Theorems to Green's Theorem (both tangential form and divergence form) in the plane. Explain that tangential form and divergence form are equivalent. Proof of the tangential form of Green's Theorem on a rectangular region.
  Lecture 26 note (v02, correction on p01).
  Lecture 27 (20220526): Section 16.4: Derive Green's Theorem in tangential form (G_T)) from Stokes' Theorem. Proof of G_T on triangular-like special region using Fundamental Theorem of Calculus. Superposition property (over adjacent domains) of G_T and G_N. How to apply G_T and G_N on non-simply connected regions (regions with holes). Section 16.5: Definition of surface integrals as limit of Riemann sum. Derivation of the formula of surface area differential dσ for parameterized surfaces and implicit surfaces.
  Lecture 27 note (v02, correction on p11).
  Homework week 15. Homework week 15 solutions.
  
• Week 16: Remote quiz 9 on 5/31 19:00.
  Lecture 28 (20220531): Section 16.5 and 16.6: Examples of surface area and surface integral. Definition of orientable surfaces. Counter example (Möbius band). Computation of n dσ in case of parametrization and case of implicit and explicit function representations.
  Lecture 28 note.
  Lecture 29 (20220602):
  Lecture 29 note.
  Homework week 16. Homework week 16 solutions.
  
• Week 17: Remote quiz 10 changed to 6/09 (Thursday) 10:10AM.
  Lecture 30 (20220607):
  Lecture 30 note (v02, correction on page 5-7).
  Homework week 17.
• Week 18: Final Exam 6/16 (Thursday) 10:10AM.
  Remote: Use recitation B05 and B08 google meet link above. Physical: B03 (the lecture room).
• New! Homework assignments (temporary) in current edition (13e): Chapter 10. Chapter 14. Chapter 15. Chapter 16.