MTH 436 is a continuation of MTH 435. We will continue studying the precise mathematical theory behind Calculus.
As in MTH 435, the emphasis will be on mathematical precision and strenthening your skills in proving theorems.
We will again study fundamental concepts of Calculus
that you already know from the intuitive point of view. This time topics include the Riemann integral, sequences of functions, series of constants, series of functions, power series, and the Taylor series.
Many theorems in analysis that we will prove are particular cases of more general theorems
in metric spaces. Therefore, we will continue our study of metric spaces.
You will find here current downloads: solutions to homework problems, additional class notes etc.
Exams and Evaluation
As in MTH 435, we will have two exams during the semester, Friday March 6, 3-5 and Friday April 17, 3-5.
The final exam will be scheduled by the Office of Enrollment Services based on the time your section meets. The schedule will be posted at:
Final Exam Schedule. According to this grid, our exam is scheduled
for Monday, May 4, 3 pm - 6 pm.
Your grade will be based upon a possible total of 500 points, as
- Exam 1 -- 100 points,
- Exam 2 -- 100 points,
- Homework -- 150 points,
- Final exam -- 200 points,
Weekly homework will be assigned in class.
We will begin with few remarks on cardinality, and a few remarks and theorems in complete metric spaces.
Then we will move to Real Analysis. After that we will study more topics in metric space topology.
Relationships between Lipschitzianity, uniform continuity and differentiabilty
Differentiability of monotone functions
Partitions, refinements, the definition of the Riemann integral
Properties of the integral
Sufficient conditions for Riemann integrability
The First Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus
Sequences of functions, pointwise convergence, uniform convergence
Uniform convergence and continuity
Convergence of integrals
Series of constants, convergence
Properties of series, absolute convergence
Alteranting series, Leibnitz Theorem
Limit inferior, limit superior
Series of functions
Convergence, uniform convergence
Series of integrals
Power series, radius of convergence
Taylor polynomials, Taylor series
Open covers, subcovers
Characterizations of compactness in metric spaces
Compact subspaces of Rn
Characterizations of continuity in metric spaces
Continuous mappings on compact spaces