A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series.
This course and MAT 319 meet together for the first 6 weeks of the semester. After the first midterm, the classes will split: students whose grade on the first midterm was higher than a certain cutoff can move to MAT 320 (or amy choose to stay in MAT319). Students whose grade was lower then the cutoff will stay in MAT 319.

- C or higher in MAT 200 or permission of instructor;
- C or higher in one of the following: MAT 203, 205, 211, 307, AMS 261, or A- or higher in MAT 127, 132, 142, or AMS 161

Tuesdays and Thursdays 10:00am - 11:20am in Light Engineering 152,

Oleg Viro

Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University

Arrived at Stony Brook in 2007.

Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University

Arrived at Stony Brook in 2007.

Office: Math Tower 5-110

Phone: (631) 632-8286

Email: oleg.viro AT math.stonybrook.edu

Web page: www.math.stonybrook.edu/~oleg

Research fields: Topology and Geometry,

especially low-dimensional topology
and real algebraic geometry.

Tuesdays and Thursdays 7:00pm - 8:00pm in Math Tower 5-110.

Tuesdays 11:30am - 12:30pm in Math Tower P-143.

Jean-Francois Arbour, PhD Student

Arrived at Stony Brook in 2015

Office: Math Tower 2-122

Email: jean-francois.arbour AT stonybrook.edu

Arrived at Stony Brook in 2015

Office: Math Tower 2-122

Email: jean-francois.arbour AT stonybrook.edu

Homework sets will be typically assigned weekly and due on Thursdays in class. They will be posted on this web site. (NOT on blackboard site!) Late homework will not be accepted. However, grades for homework assignments may be dropped in cases of documented medical problems or similar difficulties.

There will be two in-class midterms; the first midterm will be on Th, 9/29. The date of the second midterm will be announced later.

Grades will be based on the following scheme: Homework -- 20%; Midterms -- 20% each; Final Exam 40%.

Elementary Analysis: The Theory of Calculus, by Kenneth Ross, 2nd edition

**Sequences.** limit of a sequence. Limit laws for sequences and
formal proofs. Monotone and Cauchy sequences. Subsequences.
Bolzano-Weierstrass Theorem, lim inf and lim sup

**Topological and metric spaces.** Axioms in terms of neighborhoods and open sets
and their equivalence. Interior, exterior and boundary points of a set in
a topological space. Metric and metric spaces. Balls and spheres in a
metric space. Metric topology.
Topology of a subspace.

**Continuous maps.** Definition of continuous maps between topological spaces and their simplest properties.
Continuity at a point and its relation to continuity.
Sequential continuity and its relation to Continuity.
Theorems about operations with continuous functions.
Theorem on sequential continuity of composition of sequentually
continuous maps.

**Extreme Value Theorem.** Sequential compactness. Sequentially compact sets in a metric space
are bounded and closed. The converse statement for subsets of
Euclidean space. Continuous image of a sequentially compact space is
sequentually compact.

** Intermediate Value Theorem**
Connected spaces. Properties of connected sets. Connected components
of a topological space. Theorem about continuous image of a connected
space. Theorems about continuity of monotone functions.

** Uniform continuity.** Uniform continuity of a continuous function
on a closed interval.

**Series** Convergence and divergence of series, the sum.
Geometric series. Cauchy criterion for convergence of series. Tests for
convergence of series. Comparison test. Absolute convergence. Ratio and
root tests. Harmonic series and its divergence. Convergence of
$\sum\frac1{n^p}$. Integral tests. Alternating series theorem.
The Riemann rearrangement theorem about conditionally convergent series.
Invariance of the sum of a positive convergent series under
permutations of its terms.

** Power series.** Convergence radius of a power series
Uniform convergence of a sequence of functions.
Continuity of the uniform limit of a sequence of continuous
functions. Continuity of power series.
Theorem about limit of integrals and integral of the limit

**Derivatives** Definitions of derivative.
Continuity of a differentiable function.
Rules for calculation of derivatives.
Vanishing of the derivative at a local extremum of a differentiable
function, Rolle's theorem and Mean Value Theorem.
Values of the derivative and behavior of the function.

**Integrals** The Darboux integral of a bounded function. The test
for integrability. Integrability of monotonic and continuous
functions. Properties of Darboux integrals,
Intermediate Value theorem for integrals.
Fundamental theorem of Calculus.
Integration by parts and theorem on change of variable.
Term-wise differentiation and integration of power series.
Taylor series for a function.

If you have a physical, psychological, medical, or learning disability that
may impact your course work, please contact Disability Support
Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.
Students who require assistance during emergency evacuation are encouraged
to discuss their needs with their professors and Disability Support
Services. For procedures and information go to the Evacuation
Guide for People with Physical Disabilities.

Each student must pursue his or her academic goals honestly and be
personally accountable for all submitted work. Representing another
person's work as your own is always wrong. Faculty are required to report
any suspected instances of academic dishonesty to the Academic Judiciary.
For more comprehensive information on academic integrity, including
categories of academic dishonesty, please refer to the academic judiciary
website.

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.