Time and Place: TTh 9:00am–10:15am, UNIV 303

Instructor: Arshak Petrosyan

Office Hours: TTh 10:15am-11:45m, or by appointment, in MATH 836

Course Description: Credit Hours: 3.00. Real analysis in one and n-dimensional Euclidean spaces–continued from MA 44000. Topics include mappings of Euclidean spaces and their derivatives, multivariable chain rule, inverse function theorem and implicit function theorem, sets with content and integration in n dimensions, the integrability theorem, Jacobian and change of variables theorem, related topics.

Textbook:

[B] R. Bartle, The Elements of Real Analysis, Second Edition, John Wiley & Sons, New York, 1975.

Additional texts:
[R] W. Rudin, Principles of mathematical analysis, Third edition, McGraw-Hill, New York, 1976
[M] J. Munkres, Analysis on Manifolds, Addison-Wesley, 1991.

Course Outline:

  • [B], Ch. VII, §§39-42 (5-6 wks.):
    • The Derivative in $\mathbb{R}^p$
      • Partial derivatives, directional derivatives, the derivative of $\ f:\mathbb{R}^p\to \mathbb{R}^q$, the Jacobian
    • The Chain Rule and Mean Value Theorems
      • Chain Rule, Mean Value Theorem, interchange of the order of differentiation, higher derivatives, Taylor’s Theorem
    • Mapping Theorems and Implicit Functions
      • Class $C^1$, Approximation Lemma, Injective Mapping Theorem, Surjective Mapping Theorem, Open Mapping Theorem, Inver- sion Theorem, Implicit Function Theorem, Parametrization Theorem, Rank Theorem
    • Extremum Problems
      • Relative extrema, Second Derivative Test, extremum problems with constraints, Lagrange’s Theorem, inequality constraints
  • [B], Ch. VIII §§43–45 (5-6 wks.):
    • The Integral in $\mathbb{R}^p$
      • Content zero, Riemann sums and the integral, Cauchy Criterion, properties of the integral, Integrability Theorem
    • Content and the Integral
      • Sets with content, characterization of the content function, further properties of the integral, Mean Value Theorem, iterated integrals
    • Transformation of Sets and Integrals
      • Images of sets with content under $C^1$ maps, transformations by linear maps, transformations by non-linear maps, the Jacobian Theorem, Change of Variables Theorem, polar and spherical coordinates, strong form of the Change of Variables Theorem

The final 2-3 weeks will be spent on additional topics, such as integration on manifolds.

Homework: There will be weekly homework assignments to be collected through Gradescope, typically due at 11:59pm on Thur. No late homeworks will be accepted, however the lowest homework score will be dropped. For more information, see Homework page.

Exams: There will be two midterm exams and a final exam (most likely take-home exams). The exact information will be posted in the Exams page.

Grading: Your final grade will be computed by the scheme

Final Score = (3/11)ME1 + (3/11)ME2 + (3/11)FE + (2/11)HW,

where where FE, MEi, HW are the scores (in %) for Final Exam, Midterm i, Homework.

Note: If you perform better than average on both midterm exams, you will be given an option of not taking the final exam and your score will be computed by an alternative scheme (to be specified towards the end of the course).

Grade cutoffs: Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a lower percentage will be enough to get that grade.

Important dates

  • Mon, Jan 23: Last day to cancel a course assignment without it appearing on record
  • Fri, Mar 10: Last day to withdraw from a course with a W or WF grade

Academic Integrity: As a reminder, all students must comply with Purdue’s policy for academic integrity:

https://www.purdue.edu/odos/osrr/academic-integrity/

Academic integrity is one of the highest values that Purdue University holds. Individuals are encouraged to alert university officials to potential breeches of this value by either emailing integrity@purdue.edu or by calling 765-494-8778. While information may be submitted anonymously, the more information that is submitted provides the greatest opportunity for the university to investigate the concern.

Students with Disabilities: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are welcome to let me know so that we can discuss options. You are also encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone at 765-494-1247.

If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor:

https://www.purdue.edu/drc/students/course-accessibility-letter.php

Attendance Policy: This course follows Purdue’s academic regulations regarding attendance, which states that students are expected to be present for every meeting of the classes in which they are enrolled. When conflicts or absences can be anticipated, such as for many University-sponsored activities and religious observations, the student should inform the instructor of the situation as far in advance as possible. For unanticipated or emergency absences when advance notification to the instructor is not possible, the student should contact the instructor as soon as possible by email or phone. For cases that fall under the University’s excused absence regulations, the student or the student’s representative should contact or go to the Office of the Dean of Students (ODOS) website to complete appropriate forms for instructor notification. Under academic regulations, excused absences may be granted by ODOS for cases of grief/bereavement, military service, jury duty, parenting leave, or emergent medical care. Absences outside of those covered by the University’s excused class absence regulations are at the instructor’s discretion. Purdue expects each student to be responsible for class-related work missed due to an unavoidable absence. Students should contact their instructors directly to discuss the absence and opportunity to complete missed coursework. This work may be made up at the discretion of the instructor.

In cases related to COVID-19, please follow the Protect Purdue Updates for the Spring 2023 Semester.

Mental Health/Wellness Statement: Purdue University is committed to advancing the mental health and well-being of its students. If you or someone you know is feeling overwhelmed, depressed, and/or in need of mental health support, services are available. For help, such individuals should contact Counseling and Psychological Services (CAPS) at 765-494-6995 during and after hours, on weekends and holidays, or by going to the CAPS office on the second floor of the Purdue University Student Health Center (PUSH) during business hours. The CAPS website also offers resources specific to situations such as COVID-19.

Nondiscrimination Statement: Purdue University is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her potential. In pursuit of its goal of academic excellence, the University seeks to develop and nurture diversity. The University believes that diversity among its many members strengthens the institution, stimulates creativity, promotes the exchange of ideas, and enriches campus life. For more information, please see Purdue’s full Nondiscrimination Policy Statement.

Emergencies: In the event of a major campus emergency, course requirements, deadlines and grading percentages are subject to changes that may be necessitated by a revised semester calendar or other circumstances beyond the instructor’s control. Relevant changes to this course will be posted onto the course website or can be obtained by contacting the instructor via email or phone. You are expected to read your @purdue.edu email on a frequent basis.