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## Mathematical Analysis

2019-2020

### Course Objective

After this course, the student...

1. ... knows basic definitions concerning limits and continuity
(convergence, Cauchy sequence, limit, completeness, continuity, uniform
continuity) and is able to determine whether a sequence, series or
function satisfies these definitions;
2. ... knows the definition of differentiability (i.e., that a function
can be approximated by a linear one) and can determine whether a
function (and in particular, a power series) is differentiable;
3. ... knows the definition of Riemann integrability and can prove that
certain functions (in particular, polynomials, monotone and uniformly
continuous functions) are Riemann integrable;
4. ... knows the definition of basic concepts from metric topology
(metric, convergence, completeness, Banach space) and can prove that
simple examples satisfy these definitions;
5. ... knows the statement of the Banach Fixed Point Theorem, and can
apply this theorem to solve fixed point equations (in particular
integral and differential equations).

### Course Content

This course treats the rigorous mathematical theory behind Calculus:
limits, continuity, linear approximation, differentiability,
integrability, and the mutual relation between these concepts. The
mathematical tools that are necessary for formulating and proving the
essential results of Calculus are first presented in the context of real
valued sequences and real valued functions of a real variable, in such a
way that everything can later be generalised (to Y-valued functions of
variables in X, with X and Y Banach spaces). The space C[a,b] of real
valued continuous functions on an interval [a,b] will appear as the
first example of such a Banach space.

Starting point of the course are an ancient iterative scheme for solving
equations, and the fundamental properties of (the set of) real numbers.
Highlights: a fairly complete exposition of power series directly based
on a systematic algebraic approach for monomials, and an early
introduction of the Implicit Function Theorem via a contraction argument
and the Banach Fixed Point Theorem.

Topics covered:
1. Cauchy sequences, convergence, limits;
2. Completeness of the real numbers; theorem of Bolzano-Weierstrass;
3. Continuity and uniform continuity;
4. The concept of differentiability (including differentiability of
power series);
5. The concept of Riemann integrability (including Riemann integrability
of monotone and uniformly continuous functions);
6. The language of metric topology;
7. Completeness of the space C[a,b]; uniform convergence;
8. The Banach Fixed Point Theorem (with applications to integral and
differential equations, and the implicit function theorem).

### Teaching Methods

Lectures, study sessions and tutorials (2+1+2 hours per week).

You are also required to hand in a homework assignment every other week.

We expect you to dedicate in total about 10 hours per week to this
course.

### Method of Assessment

A written midterm exam [40%];
A written final exam [60%];

You will also be required to hand in 6 written assignments.

To pass the course...

... your final score must be no less then 55% (all students);
... 5 out of your 6 hand-in assignments must have been graded as
"sufficient" (all students). A hand-in assignment that is initially
graded as insufficient, may be handed in a second time;
... you must have been present at 70% of the study sessions and
tutorials (full time students only);

If you don't fulfill these requirements, then you must take the resit
exam. The resit exam then counts for 100% (i.e. the partial grades that
you obtained during the course in period 1 will no longer be valid).

### Entry Requirements

Basic Concepts in Mathematics (or another course on general mathematical
language, notation, and concepts, including proof by induction and
elementary combinatorics);

### Literature

Course notes will be made available through Canvas.

### Target Audience

Bachelor Mathematics Year 1

Participation in 70% of study sessions and tutorials is mandatory (for
full time students) in order to pass the course.

### Recommended background knowledge

Single Variable Calculus

### General Information

Course Code XB_0009 6 EC P4+5 100 English Faculty of Science prof. dr. J. Hulshof prof. dr. B.W. Rink prof. dr. J. Hulshof

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods Seminar, Lecture
Target audiences

This course is also available as: