Mathematical Analysis 2

Bachelor course in Engineering Sciences 2021/22


Course led by Oliver Butterley, in collaboration with Alessio Ranallo.

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 weeks of the course are devoted to reviewing the material already presented.

Course content

Lecture notes are available to download. They are a collaborative work in progress, constantly updated during the course. Slides from 2020/21 are still available from the archive and the material covered is identical to this year.

A list of core skills is available. This is not the entire content of the course, this is a list of the core skills and is provided as a convenience. To obtain top grades a full understanding of all the material of the course is required. Nonetheless, a good ability with the majority of skills listed here is sufficient to pass.

Part Topics Teaching period Exercises Due date
Sequences and series of functions {{ ! ? 'Show topics' : 'Hide' }}
  • Sequences / series of functions
  • Pointwise / uniform convergence
  • Power series
  • Radius of convergence
  • Differentiating / integrating
  • Taylor's series
  • Differential equations
20/09/21 - 01/10/21 Exercises 1 03/10/21
Differential calculus of scalar and vector fields {{ ! ? '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
04/10/21 - 15/10/21 Exercises 2 17/10/21
Extrema and other applications {{ ! ? '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
18/10/21 - 29/10/21 Exercises 3 31/10/21
Curves and line integrals {{ ! ? '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 conservative
  • Applications to differential equations
01/11/21 - 12/11/21 Exercises 4 14/11/21
Multiple integrals {{ ! ? '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
15/11/21 - 26/11/21 Exercises 5 28/11/21
Surfaces and surface integrals {{ ! ? 'Show topics' : 'Hide' }}
  • Parametric representation of a surface
  • Fundamental vector product
  • Area of a surface
  • Surface integral
  • Stokes' theorem
  • Curl and divergence
  • Gauss' theorem
29/11/21 - 10/12/21 Exercises 6 12/12/21


  • Completing 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. Each set of exercises is composed of 10 questions of varying difficulty (but equal points).

Practical details

  • Join the MS Team for the course using the team code: 1q0s4s2.
  • The course is taught in Semester 1 with the following schedule:
    • Mon 14:00-14:45, 15:00-15:45 in Aula 8
    • Wed 9:30-10:15, 10:30-11:15 in Aula A3
    • Fri 14:00-14:45, 15:00-15:45 in Aula C2
    Lectures will additionally be streamed.
  • 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 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, in person, using Moodle). There is no oral exam, the grade from the written exam (possibly with bonus from the problem sets) is your final grade.
  • The exam is passed if the final mark is at least 18/30.
  • Results of the exam are available, automatically on the system, immediately the exam finishes.
  • During the year there are three exams sessions and two calls in each session. The sessions are according to the university schedule.
  • Students are permitted to bring only the following items to their desk in the exam room:
    • A single A4 sheet of paper (writing permitted on 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).
  • 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.
  • Under penalty of exclusion, during written tests the use of electronic devices and applications, except those required to access to the Moodle quiz and a basic calculator, is not allowed. It is not permitted to use books and notes during the exam. Note that only basic calculators are permitted, not the programmable type, particularly not ones which can perform integration.
  • A selection of mock exam questions are available for practice. The exam is divided into five questions on different topics covering the course material, similar to the exams of previous years.
  • Students are permitted to attempt both calls available in a given exam session and take the highest grade of the two attempts.
Because of technical reasons (a minority of students finding cheating irresistible), for the remainder of this academic year, the exams will be in paper form. The style will be the same as the Moodle but the answer sheet is submitted at the end of the exam and grades published soon after.

Exam schedule

Winter session Call I Monday 17/01/2022 14:00-17:00. Aula A3 Questions and solutions
Call II Friday 31/01/2022 14:00-17:00. Aula 4 Questions and solutions
Straordinario Call X Friday 29/04/2022 14:00-17:00. (info)
Summer session Call III Tuesday 14/06/2022 14:00-17:00. Room: Aula B2. Questions and solutions
Call IV Friday 01/07/2022 14:00-17:00. Room: Aula 1. Questions and solutions
Autumn session Call V Thursday 01/09/2022 14:00-17:00. Room: Aula C1. Questions and solutions
Call VI Thursday 15/09/2022 14:00-17:00. Room: Aula B5. Questions and solutions

Lecture diary

Date Topics Reference