**MATH 402 **Advanced Calculus II - MATH 402 002 CRN6016.

Hybrid Instructional Method: Face-to-Face Plus: 2:00 pm - 3:15 pm T SMLC B81, Remote Set Day/Time 2:00 pm - 3:15 pm R Remote Instruction Microsoft Teams.

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Please note the following information. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I will send you by email a link to join the MATH402 team. Please follow it in order to request access. You can find the link to join the team in Canvas as well. After you join the class MS Team. I will drop from the class all students who do not request access to Microsoft Teams by Friday, January 27. The remote class on Thursday will be through MS Teams. __Homework, including submission, detailed topics list of the covered material and all remaining course information can be found in the class MS OneNote. This page contains only the basic information for the class, use the class MS Teams and OneNote for an up to date information throughout the semester.__

Our classroom and our university should always be spaces of mutual respect, kindness, and support, without fear of discrimination, harassment, or violence. Should you ever need assistance or have concerns about incidents that violate this principle, please access the resources available to you on campus. Please note that, because UNM faculty, TAs, and GAs are considered "responsible employees" by the Department of Education, any disclosure of gender discrimination (including sexual harassment, sexual misconduct, and sexual violence) made to a faculty member, TA, or GA must be reported by that faculty member, TA, or GA to the university's Title IX coordinator. For more information on the campus policy regarding sexual misconduct, please see: https://policy.unm.edu/university-policies/2000/2740.html.

Support: LoboRESPECT Advocacy Center and the support services listed on its website, the Women's Resource Center and the LGBTQ Resource Center all offer confidential services and reporting.

**Instructor: **Dimiter Vassilev** Office : SMLC** 326 **Email: ****vassilev@unm.edu****
Phone Number**: 505 277 2136** **

**Office
Hours**: I will have office hours through Microsoft Teams on T 10:00am-11:00am & Th 1:00pm-2:00pm. You can also send me an email to arrange a meeting or stop by my office if you have a quick question.

**Student Attendance: **__Students are expected to attend all meetings of the classes in which they are enrolled. A student with excessive absences may be dropped from a course by the instructor with a grade of W or the student may receive a grade of F at the end of the semester.__ Absences due to illness, or to authorized University activity such as field trips, athletic trips, etc., are to be reported by the student to his/her instructor(s) and to the Dean of Students Office. If a student is unable to contact his/her instructor(s) the student should leave a message at the instructor's department. The reporting of absences does not relieve the student of responsibility for missed assignments, exams, etc. The student is to take the initiative in arranging with his/her instructor(s) to make up missed work, and it is expected that the faculty member will cooperate with the student in reasonable arrangements in this regard. Please, see https://pathfinder.unm.edu/campus-policies/class-absences-and-student-attendance.html for more details.

**MATH402 Catalog Course Description**.
Generalization of 401/501 to several variables and metric spaces: sequences, limits, compactness and continuity on metric spaces; interchange of limit operations; series, power series; partial derivatives; fixed point, implicit and inverse function theorems; multiple integrals. **Prerequisite**: MATH401

**Text: **Introduction to Analysis, An, 4th edition, William Wade

**Semester Deadline Dates**: https://registrar.unm.edu/semester-deadline-dates/spring-2023.html

**Collaboration. **I encourage you to __work with your
peers and me__ on the homework, provided you __write up and submit your own
solutions in your own words__.

**Homework. **Homework
is due every Thursday at the beginning of the class in MS Teams. I encourage you to work on
the homework with your classmates, but you are __required to write up your own
solutions in your own words__. To help the grader, please write your solutions neatly using correct grammar
and mathematical notation (no points will be given for work that the reader
cannot follow). The ten best homework grades will be used in
computing the homework score. Please do not turn-in late homework! The syllabus also lists
recommended homework problems. These are NOT to be handed in. Work as
many as it takes for you to understand the material. You should see me as early and as often as necessary if you
are having difficulties with the homework problems.

**Exams**.
The exam dates are given in the schedule posted in the class OneNote. The Final Exam date is set by UNM, see https://schedule.unm.edu/final-exams/final_exam/spring2023.pdf. No makeup exams will be given
unless you contact me ahead of time with a documented “university authorized
absence”, including, but not limited to illness, family emergency, active
participation in scholarly or athletic events. Exams may include some multiple-choice questions testing very specific skills or concepts. The exams will be predominantly based on the homework and in-class problems. Students
having conflicts with the examination schedule must notify me before TBD. Any
student having more than three examinations scheduled in any one day may notify
the instructor of the last examination listed. If notified before TBA, I shall make arrangements to give a special examination. Conflicts
arising as a result of scheduling out of normal
hours-pattern or day sequences must be resolved by the instructor of the
off-pattern courses.

** Assessment (including grading). **You should think of most of
the work during the semester including homework and the midterm exams as means for feed-back and learning. This will be reflected
in the grading policy where I will drop about 25%-30% of the lowest homework. The Final Exam will be an opportunity for a
major grade change by showing a cumulative achievement of the course objectives. There will be some opportunities for bonus points.

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Support: Contact me by email or in office hours and contact Accessibility Resource Center (https://arc.unm.edu/) at arcsrvs@unm.edu (505) 277-3506.

** Topics and Schedule. Week of:**

- 1/16/23 Chapter 6 Infinite series of real numbers - tests for convergence.

- 1/23/23 Chapter 7.1 Uniform convergence of sequences & begin 7.3 Power series.

- 1/30/23 Chapter end of 7.3 Power series & 7.4 Analytic functions.

- 2/6/23 Chapters: finish Analytic functions 7.4, Metric spaces 10.1 & Limits in Euclidean space 9.1.

- 2/13/23 Convex sets & functions, Jensen's inequality, (generalized) AM-GM, Young's, Holder's and Minkowski's inequalities. Continue 10.1 & 10.2, 8.1, 8.3, 9.1.

- 2/20/23 Exam 1 on 2/21/23. Adherent points and closure of a set. Compact metric spaces, sequential compactness, total boundedness. Lindelof's theorem. Characterization of compactness. Separable spaces, dense sets. Ch's: 9.1, 9.5, 10.3, 10.4.

- 2/27/23 Subspaces, totally boundedness and subspaces, complete subspaces, compact spaces and subspaces. The Heine-Borel theorem in Euclidean space. Ch's: 9.2, 9.5
**.**Application of compactness - contraction maps and fixed points; nowhere dense sets; Baire's category theorem. Connected spaces, Chapters 8.3, 10.5.

- 3/6/23 More on connected sets, Chapters 8.3, 10.5, path-connected sets. Cluster points, limits of functions, Chapters 9.3, 10.2. Continuous functions, Chapters 9.4, 10.6.

- 3/13/23 Spring Break

- 3/20/23 Continuous functions defined on compact sets, homeomorphism theorem, uniform continuity, uniform convergence of series defined on a metric space, integrals depending on a parameter, Ch's 10.4, 10.6, 11.1. Differentiability of vector valued functions Ch's 11.1 and 11.2.

- 3/27/23 Continue Differentiability of vector valued functions Ch's 11.1, 11.2, 11.3 and 11.4: differentiability and partial derivatives, matrix and component expressions of the derivative, the chain and other rules for differentiation, continuously differentiable functions. 4/3/23 Exam 2 on April 4

- 4/10/23 The Mean Value Inequality (MVI), Derivatives of higher order, Taylor's formula, The inverse function theorem: Ch. 11.5, 11.6

- 4/17/23 Jordan regions Ch. 12.1; sets of (Lebesgue) measure zero, compact sets of Lebesgue measure zero and of (Jordan) volume zero; images of sets of measure zero under continuously differentiable functions Ch. 12.2 + notes.

- 4/24/23 The Riemann integral - definition and properties, relation to Jordan volume (content); Lebesgue's theorem Ch. 12.2 + notes.; Fubini's theorem 12.3; Change of variables Ch. 12.4.

- 5/1/23 Ch. 13 Integral formulas - surface integral over of a function over a piece-wise surface; orientable surfaces, flux of vector field; line integrals and circulation of vector field along a closed curve. The divergence theorem and the integration by parts formula; Stokes' theorem (13.6)

Final Exam, Tuesday, May 9, 10am - 12pm.

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