Mathematical Analysis 2

Bachelor course in Engineering Sciences 2020/21


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 I-11 Part 1 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 II-8 Part 2 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 II-9 Part 3 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 II-10 Part 4 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 II-11 Part 5 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 II-12 Part 6 {{ ! ? 'Show topics' : 'Hide' }}
  • 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 (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 14:00-14:45, 15:00-15:45
    • Wed 9:30-10:15, 10:30-11:15
    • Fri 14:00-14:45, 15:00-15:45,
  • 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 @
  • Office hours (online) with Oliver Butterley by appointment (
  • 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 ( prior to the day of the exam in order schedule this extended oral.

Exam schedule

Winter session Call I Written: Monday 18/01/2021 10:00-13:00, Oral: 14:00. Questions and solutions
Call II Written: Monday 15/02/2021 10:00-13:00, Oral: 14:00 Questions and solutions
Summer session Call III Written: Tuesday 22/06/2021 10:00-13:00, Oral: after written. Questions and solutions
Call IV Written: Wednesday 07/07/2021 10:00-13:00, Oral: after written. Questions and solutions
Autumn session Call V Written: Monday 13/09/2021 14:30-17:30, Room: C12 Questions and solutions
Call VI Written: Thursday 16/09/2021 14:30-17:30, Room: C12 Questions and solutions

Lecture diary

Date Topics Reference