Time and Place: MWF 12:30pm–01:20pm, REC 315

Instructor: Arshak Petrosyan

Office Hours: MWF 11:30am-12:30pm, or by appointment, in MATH 610

Course Description: Credit Hours: 3.00. Topics covered may include a unified modern treatment of functions of several variables. Topics covered include the topology of Euclidean spaces, mappings of Euclidean spaces, Riemann integration, and integration on manifolds.

Textbook:

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

[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 will be collected weekly on Wednesdays (with some exceptions), at the beginning of class. No late homeworks will be accepted, however, the lowest homework score will be dropped. The assignments will be posted here at least one week prior to the due date.

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.

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).

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