Overview
The material of the course is divided into six parts as listed below. Each part takes two weeks and is
accompanied with a set of exercises. Mathematically the parts all build on each other and are intimately
linked. The final three weeks of the course are devoted to reviewing the material already presented.
Part 
Reference 
Slides 
Topics 
Teaching period 
Exercises 
Due date 
Introduction 

Part 0 




Sequences and series of functions 
Apostol I11 
Part 1 
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 Sequences / series of functions
 Pointwise / uniform convergence
 Power series
 Radius of convergence
 Differentiating / integrating
 Taylor's series
 Differential equations

21/09/20  02/10/20 
Exercises 1

01/11/20 
Differential calculus of scalar and vector fields 
Apostol II8 
Part 2 
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 Higher dimensional space
 Open sets, limits, continuity
 Partial derivatives
 Derivatives of scalar fields
 Derivatives of vector fields
 Level sets, tangent planes
 Jacobian matrix

05/10/20  16/10/20 
Exercises 2

01/11/20 
Some application of the differential calculus 
Apostol II9 
Part 3 
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 Linear PDEs
 1D wave equation
 Extrema (minima, maxima, saddle points)
 Second order Taylor formula
 Hessian matrix
 Extrema with constraints
 Lagrange multiplier method

19/10/20  30/10/20 
Exercises 3

09/11/20 
Curves and line integrals 
Apostol II10 
Part 4 
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 Definition of paths and line integrals
 Change of parametrization
 The second fundamental theorem of calculus
 The first fundamental theorem of calculus
 Potential functions
 Sufficient condition for a vector field to be a gradient
 Applications to differential equations

02/11/20  13/11/20 
Exercises 4

23/11/20 
Multiple integrals 
Apostol II11 
Part 5 
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 Step functions and partitions of rectangles
 Definition of integrability
 Evaluation of the integral
 Applications of multiple integrals
 Green's theorem
 Change of variables
 Polar / spherical / cylindrical coordinates

16/11/20  27/11/20 
Exercises 5

07/12/20 
Surfaces and surface integrals 
Apostol II12 
Part 6 
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 Parametric representation of a surface
 Fundamental vector product
 Area of a surface
 Surface integral
 Stokes' theorem
 Curl and divergence
 Gauss' theorem

30/11/20  11/12/20 
Exercises 6 
21/12/20 
Exercises

Exercises (problem sets) are using the Moodle platform.
Unlimited attempts are permitted for each question.

Completely the exercises in a timely way (due dates listed above) gives a 10% bonus towards the
exam.

In order to qualify for this bonus you must score at least 50% in the exercise sets for each part of the
course prior to the due dates for the exercise set for that part of the course (dates listed above).
The bonus gained during a given year only applies to the exams taken during the same academic year.

Unlimited attempts are permitted for the exercises. Anyone may ask for hints and you are encouraged to
discuss the exercises and help each other understand the material. The teams
channel can be used as a forum for discussion and asking for hints.
Practical details
 All lectures are online in a MS Team (2020/21 team code: v3z480u, 2021/22 team code: 1q0s4s2).
Lecture schedule: Semester 1,
 Mon 14:0014:45, 15:0015:45
 Wed 9:3010:15, 10:3011:15
 Fri 14:0014:45, 15:0015:45,
 Suggested references:
 Tom M. Apostol, "Calculus", Volumes 1 and 2 (2nd edition)
 Terence Tao, "Analysis 1" and "Analysis 2" (3rd edition)
 Paul Dawkins (online notes and exercises)
 Walter Rudin, "Real and Complex Analysis"

MA2
@ didattica.uniroma2.it
 Office hours (online) with Oliver Butterley by appointment (butterley@mat.uniroma2.it).

Course material from previous years and other instructors is available
Exam rules
 The exam consists of a written test (3hr, using Moodle)
and a
compulsory oral test, both online. These arrangements are subject to possible
variation if university regulations change.
 To access the oral, a grade of at least 18/30 must be obtained on the written test.
The exam is
passed if the final mark is at least 18/30.
 The written test and the oral test must take place in the same call.
 During the year there are three exams sessions and two calls in each session. The sessions are according
to the university schedule.
 Under penalty of exclusion, during written tests the use of electronic devices and applications except
those required (MS Teams and possibly a web browser to access to the Moodle quiz) is not allowed. It is
not permitted to use books and notes.
 It is necessary to have a webcam and a microphone to take the test.

A selection of mock exam
questions is available for practice. The exam is divided into five questions on different topics
covering the course material, similar to the exams of last year.
Call V & Call 6
These exams will be held in person (offline) with the following rules:
 The format of the exams is a 'Moodle test', in the same style and same topics as the previous calls this
year.
 Students are permitted to bring only the following items to their desk in the exam room:

A single A4 sheet of paper (handwritten, both sides) with whatever course notes are wanted.

Pens and pencils.

A single device (tablet / laptop) for accessing the electronic test.

An identity document.

Green pass QR code (paper or electronic).

Results of the exam are available, automatically on the system, immediately the exam finishes.

There is no oral exam, the grade from the written exam is your final grade

Paper for rough calculations will be provided in the exam room. After the exam the paper used during the
exam remains in the exam room.

Students can choose to not use the electronic test and submit the answers on paper (in this case the grade
will be available soon after the exam finishes but not immediately).

During the exam it is forbidden to communicate, using any means, with anyone except the exam invigilators.
All messaging apps must be deactivated on the devices used for the test.

Students are required to arrive at the exam room before the scheduled start of the exam.

If, for some reason a student is not able to attend the exam in person, then an online option will be
arranged. In this case the student does not do a written exam. Instead there will be an extended oral
during which will test the student's knowledge and ability with the material of the course. The topics
tested are just the same as in the written exam. In order to arrange this the student must communicate
this requirement (butterley@mat.uniroma2.it) prior to the day of the exam in order schedule this extended
oral.
Exam schedule
Lecture diary
Date 
Topics 
Reference 
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