Topics in Real Analysis
3
In person
TTh 15:30-16:50
Wachman 306
Wachman 1020
Mondays (virtual), Tuesdays and Thursdays (in person): 1:30 - 2:30 PM, and by appointment.
- We shall start the course with material regarding Fourier Series from your last semester's textbook Understanding Analysis by Stephen Abbott - available for free at https://link-springer-com.libproxy.temple.edu/book/10.1007/978-1-4939-2712-8
- The textbook for the remainder of the course is ADVANCED CALCULUS, Second Edition, by Gerald B. Folland - available for free at chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://sites.math.washington.edu//~folland/AdvCalc24.pdf
The course grading will be based on the following scheme:
Midterms count 20% each; the final counts 20%; the homework counts 20%; student final oral presentations count for 20%.
is A: 90-100; B: 80-89; C: 70-79; D 60-69; F: <60.
This is the second course in real analysis. Topics include point-set topology of Euclidean spaces, differentiation and integration of functions of several variables, Fourier series, and the Lebesgue integral (time permitting).
Course goals include:
- develop mathematical maturity and precision through transitioning from computational calculus to abstract reasoning
- further develop proficiency in mathematical proofs: write clear, correct and complete proofs using definitions and theorems and critically assess the validity of analytical arguments
- connect theory with applications - study fundamental analytical tools used to solve problems in geometry, optimization, and applied mathematics
- use Fourier Analysis on the real line to study convergence, approximation and frequency behavior of functions of a single variable
- use rigorously the structure of Euclidean Spaces: understand completeness, compactness, and connectedness, and apply these concepts in proofs
- formulate and prove results concerning limits, continuity, and uniform continuity for functions of several variables
- understand differentiation of functions of several variables: work with total derivatives, gradients, Jacobians, and correctly utilize the Chain Rule, the Mean Value Theorem and Taylor's Theorem
- understand the local analysis of functions of several variables through fundamental results such as the Inverse and Implicit Function Theorems
- understand integration in several variables through the study of the Riemann Integral, and change of variables
The course starts by introducing elements of Fourier analysis on the real line, including Fourier series, the Fourier transform, highlighting convergence, approximation, and applications to analysis and differential equations. After this the focus shifts to analysis of functions of several real variables. Topics include the structure of euclidean spaces, limits, continuity, compactness, and convergence, differentiation and integration in several variables, including major theorems such as the Mean Value Theorem, Taylor’s Theorem, and the Inverse and Implicit Function Theorems, with applications to optimization and geometry.
Midterm 1: in class, Thursday, February 26. Midterm 2: in class, Thursday April 16. Final Exam: Tuesday May 5th, 1:00 pm - 3:00 pm
Attendance directly correlates to a good performance in this course! Since a critical aspect of your learning is engaging with the class, students are expected to attend lectures as active participants. If you have an excuse for missing a class, please let me know. If you have 4 or more unexcused absences, your grade will drop half a notch (e.g. B to B-) for each 4 unexcused absences.
A student absent from class bears full responsibility for all material discussed in class. If you do have to miss a class, get the notes from another student and read the relevant material from the textbook.
Homework is due at the priorly announced dates/times. Each student has three flex days for the semester. These days allow you to submit a homework assignment up to three days late without penalty. You can use these days for any assignment and for any reason. You do not need to provide me with the reason: simply email me and tell me how many of your flex days you would like to use. Once you've exhausted your flex days, then point deductions will occur for any assignment submitted after the deadline.
- An assignment submitted within 72 hours of the due date will only be eligible for 75% of the maximum number of point allotted.
- An assignment submitted between 72 hours and 120 hours after the deadline will only be eligible for 50% of the maximum number of point allotted.
- Assignments submitted more than a week after the due date will not receive any credit or feedback.
If you experience extenuating circumstances prohibiting you from submitting your assignments on time, please let me know as soon as possible. I will evaluate these instances on a case-by-case basis.
This is a registered Canvas course (you can login to Canvas at https://canvas.temple.edu). Please check Canvas daily for course announcements.
There will be no regularly scheduled make up exams. Students may be excused from a test only for medical or other extreme situations, which require a valid documentation, and when possible, advance notification. In the case that you are unable to take an exam as scheduled, please contact me as soon as possible to make alternate arrangements.
To achieve course learning goals, students must attend and participate in classes, according to the course requirements. However, if you have tested positive for or are experiencing symptoms of a contagious illness, you should not come to campus or attend in-person classes or activities. It is the student’s responsibility to contact me to create a plan for participation and engagement in the course as soon as you are able to do so, and to make a plan to complete all assignments in a timely fashion.
It is important to foster a respectful and productive learning environment that includes all students in our diverse community of learners. Our differences, some of which are outlined in the University's nondiscrimination statement, will add richness to this learning experience. Therefore, all opinions and experiences, no matter how different or controversial they may be perceived, must be respected in the tolerant spirit of academic discourse. Remember to be careful with your own and others’ privacy. In general, have your behavior mirror how you would like to be treated by others.
Students are expected to engage respectfully and professionally in all aspects of the course. Active participation, thoughtful questions, and constructive discussion are encouraged, while maintaining an environment that supports focused learning. Academic integrity is essential: all submitted work must reflect the student’s own understanding, and collaboration should follow the guidelines stated for each assignment. Punctuality, preparedness, and consideration for others are expected at all times, both in and out of class.
Any student who has a need for accommodations based on the impact of a documented disability or medical condition should contact Disability Resources and Services (DRS) in Howard Gittis Student Center South, Rm 420 (drs@temple.edu; 215-204-1280) to request accommodations and learn more about the resources available to you. If you have a DRS accommodation letter to share with me, or you would like to discuss your accommodations, please contact me as soon as practical. I will work with you and with DRS to coordinate reasonable accommodations for all students with documented disabilities. All discussions related to your accommodations will be confidential.
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The use of generative AI tools (such as ChatGPT, DALL-E, etc.) is not permitted in this class unless specifically announced for a particular assignment; therefore, any use of AI tools for work in this class may be considered a violation of Temple University's Academic Honesty policy and Student Conduct Code, since the work is not your own. The use of unauthorized AI tools will result in a grade of zero on the assignment; a second offense will be reported to the Student Conduct Board.
The grade "I" (an "incomplete") is only given if students cannot complete the course work due to circumstances beyond their control. It is necessary for the student to have completed the majority of the course work with a passing average and to sign an incomplete contract which clearly states what is left for the student to do and the deadline by which the work must be completed. The incomplete contract must also include a default grade that will be used in case the "I" grade is not resolved by the agreed deadline. See the full policy by clicking here (opens in new tab/window).
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