Math 442

MATH 442

COURSE INFORMATION

Scheduled: MWF 12:30-1:20pm in Rec 112
Instructor: Róbert Szőke
email rszoke@math.purdue.edu
Office hours:TTh 1:00-2:00pm, or by appointment, in MATH 718
Grader: Michael Perlmutter

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

Textbook:
[B] Robert G. Bartle: The elements of real analysis (Wiley, 2nd ed., 1975)

Additional text:

Course Outline:

[B], Ch. II, Topology of R^p: Heine-Borel, connectedness, etc.
[B], Ch. III, §§14-17 : Sequences, Bolzano-Weierstrass thm., Cauchy criterion.
[B], Ch. IV, §§20-22: Continuity
[B], Ch. VII, §§39-41: Differentiation, mapping theorems.
[B], Ch. VIII: Riemann integration, including "content", Lebesgue's criterion for integrability, and change of variables.

Here is a correct proof of 42.13Corollary from p407:
Proof

Exams:

Midterm 1: Friday, Feb 22, 12:30pm-1:20pm in class (Rec 112).
solutions
Test is going to be over the topics covered in the period Jan 07- Feb 11 (see the Course Log).
Practice problems for Midterm1

Midterm 2: Wednesday, April 10, 12:30pm-1:20pm in class (Rec 112).
solutions

Test is going to be over the topics covered in the period Feb 13- March 29 (see the Course Log).
Practice problems for Midterm2

FINAL EXAM: Monday, April 29, 8:00am-10:00am, Rec 112
solutions You can take a look at the exam today (April 30th) in my office till 15:00pm.

The final exam basically covers everything we did in this semester (except the change of variables theorem).
Practice problems for the final

Grading: Your course grade will be determined using the following distributions:
Homeworks 200 pts
2 midterms, 100 pts each
Final exam 200 pts

Homeworks:
Homework will be collected weekly on Fridays. The assignments will be posted on this website at least one week prior the due date.
#1 (due JAN 18, in class): 8Q, 8beta(a,b,c), 9G, 9H, 9I, 9L, 10C, 10F
#2 (due JAN 25, in class): 11H, 11N, 11P, 12C, 12I
#3 (due FEB 1, in class): 16Q, 20K, 20P, 21L
#4 (due FEB 8, in class): 22F, 22G, 22H, 22S, 39D, 39E, 39J
#5 (due FEB 15, in class): 39T, 39V, 39W, 40E, 40H, 40J, 40K, 40R
#6 (due MARCH 1, in class):40S, 40T, 40U, 41A, 41D, 41G, 41J, 41L
#7 (due MARCH 8, in class): 41K, 41U, 41V(can argue directly, you do not need to follow the book's suggestion to use the proof of thm 41.6), 41W, 41Na-b, 41O,
#8 (due March 22 in class): 42A(a-c), 42D, 42F(d,e), 42P, 42Q, 42U
#9 (due March 29 in class): 42R, 42S(c), 43B,43D, 43F
#10 (due April 05 in class): 43H, 43I, 43M, 43Va-b-c-d
#11 (due April 22 MONDAY!, in class): 44D, 44G, 44J, 44Q, 44P, 44R. 44S,
Course Log:
Covered in class:
Jan 07: Sect.8:Cartesian spaces, inner products, norms
Jan 09: Sect.9:Open and closed sets, interior, boundary, closure
Jan 11: Sect.10 and part of Sec.11:Cluster points, Bolzano-Weierstrass, Nested Cells thm,compact sets
Jan 14: Sect.11: Heine-Borel, Cantor Intersection, Lebesgue covering Thm.
Jan 16: Sect. 11 cont:Nearest point thm, Sect. 12: connected sets
Jan 18: Sect. 14 Convergence in R^p spaces, Sect. 16 Bolzano-Weierstrass thm for sequences,
Jan 21: No class. MLK day
Jan 23: Sect. 16. cont.:Cauchy sequences, Sect. 20: Continuity in a point, characterizations of continuity, examples
Jan 25: Sect.21: Linear maps, Sect. 22:Global continuity
Jan 28: Sect. 22: cont maps preserves connectivity and compactness. Corollaries.
Jan 30: Sect. 22: continuity of the inverse, Sect. 39: partial derivative, limits, differentiability, examples, uniqueness of the derivate
Feb 1: Sect 39:Differentiability implies continuity, partial derivatives of maps wrt any vector, differentiability of maps is equivalent to the differentiability of the coordinate functions, Jacobian matrix
Feb 4: Sect.39: Thm on the existence of the derivative,tangent plane, Sect40. combinations of differentiable functions
Feb 6: Sect.40.:combinations of differentiable functions continued,chain rule
Feb 8: Sect.40: chain rule for partial derivatives, mean value theorem (scalar and vector valued version)
Feb 11:Sect.40:interchange of the order of differentiation, higher order derivatives
Feb 13:Sect.40: Taylor's thm, Sect41:C^1 functions, a mean value thm
Feb 15:Sect.41:approximation lemma, injective mapping thm, Banach fixed point thm
Feb 18:Sect.41: surjective mapping thm, open mapping thm, inverse mapping thm (started)
Feb 20:review for midterm1
Feb 22: Midterm1
Feb 25: Sect.41:inverse mapping thm finished, linearization of nonlinear problems, example to illustrate the surjective mapping thm, implicit function thm started
Feb 27: Sect.41:implicit function thm in 2 variables,block partial derivatives
March 1: Sect.41:implicit function thm, Sect42:extremum points
March 4: Sect.42: extremum problems, second derivative test, positive (neg) definite matrices
March 6: Sect.42: second derivative test finished, extremum problems with one constraint, example
March 8: Sect.42: more examples, extremum problems with multiple constraints, appl:symmetric matrices have an orthonormal bases of eigenvectors (started)
March 18: appl:symmetric matrices have an orthonormal bases of eigenvectors (finished), Sect.42 geometric explanation for Lagrange'thm, constraints with inequalities (started)
March 20: Sect.42 constraints with inequalities (finished)
March 22: Sect.43: cells, content zero, partitions
March 25: Sect.43:definition of the integral, Riemann-, upper-, lower sum, upper and lower integral
March 27: Sect 43: Riemann's criteria of integrability, equivalent characterizations of integrability
March 29: Thm on uniform continuity on compact sets in R^p, continuous functions on a cell are integrable, integrability on general sets, properties of the integral.
April 1:Sect.43: Existence of the integral (thm 43.8, thm 43.9), sets with content zero:Cantor set, Sierpinski carpet, Sierpinski gasket, Monge spounge
April 3:The problem of measure. Sect44.: sets with content, properties of the content function (started)
April 5: properties of the content function (finished), characterization of the content function (started)
April 8: review for midterm2
April 10: midterm2
April 12: characterization of the content function (finished)
April 15: Sect44: Further properties of the integral, mean value thm.
April 17: Fubini's theorem for continuous functions, iterated integrals, integration over normal domains
April 19: General Fubini theorem for integrable functions, Cavalieri principle, oscillation of functions
April 22: sets of measure zero, reviewed, Lebesgue's criteria of integrability
April 24: Change of variables theorem, examples, an open set without content
April 26:review for the final
April 29: Final exam 8:00am-10:00am (Rec 112