# MA44200 Multivariate Analysis I (Honors)

Purdue University Spring 2009

## Sunday, April 19, 2009

### Midterm Exam 2

Scheduled Tue, Apr 21, 8:00 - 9:30pm in MATH 211.
Test is going to be over the topics covered in the period Feb 23 – Apr 13 (see the Course Log)

[Practice problems]

## Saturday, April 18, 2009

### Homework

#10 (Due on Fri, Apr 17):   44.K, 44.O, 44.P, 44.R, 45.B, 45.C
[Solutions]
#9 (Due on Fri, Apr 10):   43.B, 43.M, 43.Q, 43.R, 44.H, 44.J
[Solutions]
#8 (Due on Wed, Apr 1):  42.A(a-c), 42.D, 42.F(d,e), 42.P, 42.Q, 42.R, 42.S(c), 42.U
[Solutions]
#7 (Due on Wed, Mar 11):  41.K, 41.N(a-b), 41.O, 41.R, 41.U, 41.V, 41.W
[Solutions]
#6(Due on Fri, Feb 27):  40.K, 40.R, 40.S, 40.T, 40.U, 41.D, 41.J
[Solutions]
#5(Due on Fri, Feb 20):  39.D, 39.G, 39.J, 39.T, 39.V, 39.W, 40.E, 40.L
[Solutions]
#4(Due on Fri, Feb 13):  20.K, 20.P, 21.L, 21.M, 22.F, 22.G, 22.H, 22.S
[Solutions]
#3(Due on Fri, Feb 6): 14.D, 15.I, 15.N, 15.O, 16.Q, 17.D, 17.I, 17.S
[Solutions]
#2(Due on Fri, Jan 30): 11.C, 11.G, 11.N, 12.C, 12.E, 12.I
[Solutions]
#1(Due on Fri, Jan 23):  8.Q, 8.beta(a-c), 9.G, 9.H, 9.I, 9.L, 10.C, 10.F
[Solutions]

## Friday, April 10, 2009

### Course Log

Planned
Wed, Apr 29: Review for Final Exam
Mon, Apr 27: §45 Problems on Change of Variables
Fri, Apr 24: §45 Jacobian Theorem, Change of Variables
Wed, Apr 22: Overview of Midterm Exam 2
Mon, Apr 20: Review for Midterm Exam 2
Fri, Apr 17: § 45 Linear Change of Variables, Transformations Close to Linear
Wed, Apr 15: §45 Transformation of Sets, Content and Linear Mappings
Mon, Apr 13: §44 Integral as Iterated Integral; §45 Transformations of Sets of Content Zero
Covered
Fri, Apr 10: §44 Further Properties of Integral, Mean Value Theorem
Wed, Apr 8: §44 Characterization of the content function
Mon, Apr 6: §44 Sets with content
Fri, Apr 3: §43 Properties of Integral, Existence of Integral
Wed, Apr 1: §43 Definition of Integral, Riemman, Upper and Lower Sums
Mon, Mar 30: §43 Content zero, cells, partitions
Fri, Mar 27: §42 Inequality Constraints
Wed, Mar 25: §42 Extremum Problems with Constraints. Examples.
Mon, Mar 23: §42 Extremum Problems: Examples; Extremum Problems with Constraints.
Mon, Mar 16 – Fri, Mar 20: Spring break
Fri, Mar 13: Class cancelled (because of evening exam)
Wed, Mar 11: §42 Extremum Problems, Second Derivative Test
Mon, Mar 9: §41 Implicit Function Theorem (continued)
Fri, Mar 6: §41 Implicit Function Theorem
Wed, Mar 4: Overview of Midterm Exam
Mon, Mar 2: Review for Midterm Exam
Fri, Feb 27: §41 Inverse Mapping Theorem
Wed, Feb 25: §41 Surjective Mapping Theorem, Open Mapping Theorem
Mon, Feb 23: §41 Injective Mapping Theorem, Surjective Mapping Theorem (started)
Fri, Feb 20: §41 Taylor's theorem, C1 functions, Approximation Lemma
Wed, Feb 18: §40 Mixed derivatives (finished), higher derivatives
Mon, Feb 16: §40 Mean Value Theorem, mixed derivatives (stared)
Fri, Feb 13: §39 Tangent planes, §40 Combinations of Diff. Functions, the Chain Rule
Wed, Feb 11: §39 Examples, Existence of the derivative
Mon, Feb 9: §39 Partial derivatives, differentiability
Fri, Feb 6: §22 Preservation of connectedness, Continuity of Inverse Function; start § 39
Wed, Feb 4: §22 Relative topology, global continuity, Global continuity theorem, preservation of connectedness
Mon, Feb 2: §20 Continuity at a point (different definitions), §21 Linear functions
Fri, Jan 30: §17 Sequences of functions, pointwise and uniform convergence
Wed, Jan 28: §§15-16 Subsequences, Bolzano-Weierstrass (revisited), Cauchy sequences
Mon, Jan 26: Finish §12; §§14 Convergence of sequences
Fri, Jan 23: §12 Connected sets
Wed, Jan 21: §11 Compactness, Heine-Borel, Cantor Intersection Theorem
Mon, Jan 19: MLK day, no class
Fri, Jan 16: §10 Cluster points, Bolzano-Weierstrass, Nested Cells
Wed, Jan 14: §9 Open and closed sets, interior, boundary, closure
Mon, Jan 12: §8 Cartesian spaces, inner products, norms

## Sunday, March 1, 2009

### Midterm Exam 1

Scheduled Tue, March 3, 8:00 - 9:30pm in MATH 211.
Test is going to be over the topics covered in the period Jan 12 – Feb 20 (see the Course Log)

[Practice Problems]

## Sunday, January 11, 2009

### Course Information

Scheduled: MWF 11:30am-12:20pm in MATH 215

Instructor: Arshak Petrosyan
Office Hours: MWF 10:30 -11:30am, or by appointment, in MATH 610

Course Description: MA44200 covers the foundations of real analysis in several variables, assuming the single variable notions of these concepts.
Prerequisite: MA44000

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

Course Outline:
[B], Ch. II, (2 wks.): Topology of Rp: Heine-Borel, connectedness, etc.
[B], Ch. III, §§14-17 (1 wk): Sequences, Bolzano-Weierstrass thm., Cauchy criterion.
[B], Ch. IV, §§20-22 (1 wks): Continuity (with emphasis on the equivalence of different definitions).
[B], Ch. VII, §§39-41 (5 wks.): Differentiation, mapping theorems.
[B], Ch. VIII (4 wks.): Riemann integration, including "content", Lebesgue's criterion for integrability, and careful treatment of change of variables.

The final two weeks will be spent on [R], Ch. 10: Differential forms and Stoke's Theorem.

Homework will be collected weekly on Fridays. The assignments will be posted on this website at least one week prior the due date.

Exams: There well be two in-class midterm exams and a comprehensive final exam (covering all topics). The exact time and place will be specified in due course.