MATH 312 Advanced Calculus II (Winter 2023)



Instructor: Yevgeniy Kovchegov
e-mail: kovchegy @math. oregonstate.edu
Office: Kidder 60
Office Phone No: 7-1379
Office Hours: TR 1:25pm - 1:55pm and 3:25pm - 3:55pm



TA: Sahir Gill
TA's office hours: M 12:00pm - 2:00pm and F 12:00pm - 1:00pm in Kidder 314




GRADING:
Homework 60%
Midterm 20%
Final Exam 20%

Place and time: TR 2:00pm - 3:20pm in SHEP 101

Textbook: Stephen Abbott, Understanding Analysis (Second Edition)

Learning goals: The goal of the course is to promote understanding, expertise, and experience in advanced calculus. In particular, the successful student will be able to prove basic results about sequences of functions, pointwise and uniform convergence, series of functions, integration, improper integral. Additional topics may include introductory aspects of multivariable calculus.

Prerequisites: MATH 311 or instructor's approval.

Syllabus:  .pdf



Schedule:
Tuesday, January 10  Pointwise convergence of sequences if functions. Uniform convergence of sequences of functions. Reading: sections 6.1 and 6.2   Lectures 1-3 slides (PDF)
Thursday, January 12  Uniform convergence of sequences of functions. Cauchy Criterion for Uniform Convergence. Continuous limit theorem. Reading: sections 6.2 and 6.3   Lectures 1-3 slides (PDF)
Tuesday, January 17  Differentiable limit theorem. Series of functions. Reading: sections 6.3 and 6.4   Lectures 1-3 slides (PDF)
Thursday, January 19  Series of functions. Power series. Reading: sections 6.4 and 6.5   Lectures 4-7 slides (PDF)
Tuesday, January 24  Limit infimum and limit supremum. Reading: Exercise 2.4.7 on p.61   Lectures 4-7 slides (PDF)
Thursday, January 26  Power series. Radius of convergence. Reading: section 6.5   Lectures 4-7 slides (PDF)
Tuesday, January 30  Review of series. Power series. Reading: section 6.5   Lectures 4-7 slides (PDF)
Thursday, February 2  Taylor series. Error in approximation. Reading: section 6.6   Lectures 8-9 slides (PDF)
Thursday, February 9  Taylor series. Lagrange's Remainder Theorem. Integral form of the remainder. Reading: section 6.6   Lectures 8-9 slides (PDF)
Tuesday, February 14  The Riemann integral. Riemann integrability. Reading: section 7.2   Lectures 10-14 slides (PDF)
Thursday, February 16  Properties of the Riemann integral. The Fundamental Theorem of Calculus. Reading: sections 7.4 and 7.5   Lectures 10-14 slides (PDF)
Tuesday, February 21  Properties of the Riemann integral. The Fundamental Theorem of Calculus. The Mean Value Theorem for definite integrals. The Integrable Limit Theorem. Reading: sections 7.4 and 7.5   Lectures 10-14 slides (PDF)
Thursday, February 23  Properties of the Riemann integral. The Fundamental Theorem of Calculus. Reading: sections 7.4 and 7.5   Lectures 10-14 slides (PDF)
Tuesday, February 28  Integrating functions with discontinuities. Improper Riemann integrals. Reading: sections 7.3 and 8.4   Lectures 10-14 slides (PDF)
Thursday, March 2  Riemann integrability. Topics: Lebesgue measure on real line. Reading: section 7.6   Lectures 15-17 slides (PDF)
Tuesday, March 7  Topics: Lebesgue measure and Lebesgue integral on real line. Reading: section 7.6   Lectures 15-17 slides (PDF)
Thursday, March 9  Lebesgue's criterion for Riemann integrability. Topics: Lebesgue measure and Lebesgue integral on real line. Reading: section 7.6   Lectures 15-17 slides (PDF)
Thursday, March 16  Q&A and review.