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 (team joining code: v3z480u). 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.

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.
Call IV Written: Wednesday 07/07/2021 10:00-13:00, Oral: after written.
Autumn session Call V Within period 23/08/2021 – 18/09/2021, dates announced later
Call VI

Lecture diary

Date Topics Reference