18798-MATH 511 001 Intro To Analysis II. 4538  Time MW 1300-1415, Location SMLC-124

 

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Instructor: Dimiter Vassilev     Office : SMLC 326  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: M&W 9:30am-10:30am & 2:30pm-3pm but feel free to stop-by anytime you have a quick question.

 

Final Exam: Friday May 12, 9am-11am in SMLC-124 (usual class room)

Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, March 31, 2017. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before Friday, March 31, 2017, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses.        

 

 

Texts: class notes based mainly on the following optional texts. See UNMLearn for course material and homework.

  • A. Knapp, Basic Real Analysis, 2nd Digital Edition;

  • W. Rudin, Principles of Mathematical  Analysis 3rd ed.;

  • V. Zorich, Mathematical Analysis I 2nd ed. (2015) and Mathematical Analysis II 2nd ed. (2016).

 

Additional standard books of interest

  • J. Munkres, Analysis on Manifolds;

  • M. Spivak, Calculus On Manifolds.

 

Please note the following guidelines for the course:

 

Course content: Continuation of 510. Differentiation in Rn. Inverse and implicit function theorems, integration in Rn, differential forms and Stokes theorem.
This is also the course that prepares graduate students for the Real Analysis Qualifying.

 

Grades: The final grade will be determined by homework (25%), two midterms( 50%) and a final exam (25%). The Final Exam score will replace all midterm scores that are lower than the Final Exam score.  All grades will be posted on UNMLearn.

 

Homework  There will be one weekly homework. You can work together on the homework, but you do need to write up your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly (no points will be given for work that the reader cannot follow- this is true also for exams), and staple the sheets.   The two lowest homework grades will be dropped. Please no late homework!  Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the course you will have built a folder with solutions to most of those problems for future reference. You should see me as early and as often as necessary if you are having difficulties with the homework problems.

 

 

 

Day Class Date

Topics                                                                                 

Due Homework (turn in following Monday)
M 1/16

 MLK - no class

 HW1 see UNMLearn
W 1/18

Linear maps on normed spaces - the operator norm; equivalence of boundedness and continuity; Banach spaces -  normally (or absolute) convergent series.
   

 
M 1/23

Compact metric space. Finite dimensional normed spaces - compactness of the unit sphere,equivalent  norms.

 HW2
W 1/25
   

 
M 1/30

Differentiation - partial derivatives, differentiability, the chain rule. Exponential of a matrix. Partition of unity (continuous version)

 HW3
W 2/1
   

 
M 2/6

 Examples of differentiable functions involving matrices; sufficient condition for differentiability (Ck functions); equality of mixed derivatives for smooth functions (Clairaut's theorem); proof of thechain rule.

 HW4
W 2/8
   

 
M 2/13

The MVT. The inverse function theorem.

 HW5
W 2/15
   

 
M 2/20

The implict function and rank theorems. Global invertibility (Hadamard, Caccioppoli and metric version theorems) and proper maps, star-shaped and convex spaces. Connectedness in topological spaces.

 HW6
W 2/22
   

 
M 2/27

The open (convex) cone of the positive definite symmetric matrices Sym+(n). The square root as a smooth map on Sym+(n). Smooth covering spaces. Taylor's formula.

Integrals depending on a parameter - continuity and differentition (with absolute or conditional convergence).

 HW7
W 3/1
   

 
M 3/6

 

 

Exam 1

 

 

Exam 1 Solutions
W 3/8
   

 
M 3/13

Spring Break

  
W 3/15

Spring Break
   

 
M 3/20

Sets of (Lebesgue) measure zero; sets of (Jordan) content zero. Oscillation of a function, Riemann-Lebesgue' theorem on Riemann integrability. Non-negative functions with vanishing integrals, functions vanishing a.e. - (non-)integrability, value of the integral when integrable. Images of sets of measure/ content zero under a Lipschitz map. Sard's theorem. Fubini's theorem.

 HW8
W 3/22
   

 
M 3/27

Proof of Fubini's theorem. Integrability over bounded subsets- Jordan measurable sets, Jordan content and the integral, properties. Improper integrals.

 HW9
W 3/29
   

 
M 4/3

Absolute convergence of improper integrals. The bounded and dominated convergence theorems. Fundtions defined by an integral - continuity and differentiation.

The change of coordinates.formula - some key steps and their proofs. Smooth partition of unity.

 HW10
W 4/5
   

 
M 4/10

Smooth partition of unity-smooth bump functions etc.

Multilinear maps, tensor products, alternating forms, the wedge product.

 HW11
W 4/12
   

 
M 4/17

Differential forms - the wedge prduct, pullbacks, the integral of a 1-form. Derivations of the algebra - the exterior derivative,

Lie derivative, interior product, Cartan's formula. Closed and exact forms, proof of Poincare's lemma in a star-shaped domain..

 HW12
W 4/19
   

 
M 4/24

Monday: Exam 2

Submanifolds of Rn - various chracterizations, boundary of a manifold.

 
Exam 2 Solutions

HW13 (last due homework)
W 4/26
   

 
M 5/1 Tangent space. Differential forms. Partition of unity. Integration of diferential forms. Stokes' theorem.

 

 HW14 (not due)
W 5/3

F

 TBA

Final Exam, Friday May 12, 9am-11am in SMLC-124 (usual class room)