Differential Calculus in Several Dimensions

Course - second cycle - 7.5 credits

Syllabus for students autumn 2020, autumn 2019

Course Code:
MA621A revision 1.1
Swedish name:
Differentialkalkyl i flera dimensioner
Level of specialisation
Main fields of study:
No main fields
Date of ratification:
16 August 2018
Decision-making body:
Faculty of Technology and Society
Enforcement date:
02 September 2019
Replaces Syllabus ratified:
30 March 2017

Entry requirements

1. Degree of Engineering in Mechanical Engineering or a degree in a related field. All degrees must be equivalent to at least 180 higher education credits.

2. At least 22.5 credits of Mathematics.

3. The equivalent of English B in Swedish secondary school or equivalent

Specialisation and progression relative to the degree regulations

This course is included in Materials Science: Master Programme (two years).


The course objective is for the student to gain an overview of theory and methods within the area of multivariable analysis targeting optimisation problems — the foundations of modelling used to solve the problems that emerge within materials science.


Part I: Functions of several variables

• Differential calculus for functions of several variables
• Optimisation problems
• Vector analysis

Part II: Introduction to partial differential equations

• Linear partial differential equations of first order
• General introduction to initial and boundary value problems, classification of linear partial differential equations of second order
• Wave equation: unbounded and bounded string (Fourier’s method), variable separation, Duhamel’s principle, energy and unicity
• Heat transfer equation: separation of variables, Duhamel’s principle, energy and unicity

Learning outcomes

Knowledge and understanding
Once the course is completed, the student shall demonstrate knowledge of:
• optimisation theory, both local and global;
• methods in vector analysis and an understanding of their interpretation when applied;
• methods to solve differential equations of first order;
• the concept of initial conditions, boundary conditions and the well-posedness of problems;
• classification of linear partial differential equations of second order; and
• mathematical models for phenomena within wave propagation and heat transfer, and understanding of the structure of solutions to these problems.

Skills and abilities
Once the course is completed, the student shall:
• demonstrate the ability to carry out calculations using and handling fundamental functions of several variables and the functions’ derivatives and integrals;
• demonstrate the ability to find solutions to simple optimisation problems;
• demonstrate the ability to solve certain types of partial differential equations, such as linear partial differential equations of first order, wave and heat transfer equations, and understand the properties of the solution; and
• demonstrate the ability to apply spectral methods (Fourier) and source function methods (Green) to solve problems within wave propagation and heat transfer for simple geometries.

Judgement and approach
Once the course is completed, the student shall:
• demonstrate the ability to choose appropriate methods to solve partial differential equations of first order and wave and heat transfer equations.

Learning activities

Lectures, exercises and seminars (approximately 40 hours) and independent study (approximately 160 hours).


Requirements for pass (grade A-E): Passed exam (4 credits) and passed written assignments (3.5 credits).
The final grade is based on the exam.

Grading system

Excellent (A), Very Good (B), Good (C), Satisfactory (D), Pass (E) or Fail (U).

Course literature and other teaching materials

Recommended literature
  • Adams, Robert A. & Essex, Christopher (2009). Calculus a complete course, Pearson Education
  • Pinchover, Yehuda & Rubinsteien, Jacob (2013). An introduction to Partial Differential Equations, Cambridge University Press

Course evaluation

The University provides students who are taking or have completed a course with the opportunity to share their experiences of and opinions about the course in the form of a course evaluation that is arranged by the University. The University compiles the course evaluations and notifies the results and any decisions regarding actions brought about by the course evaluations. The results shall be kept available for the students. (HF 1:14).

Interim rules

When a course is no longer given, or the contents have been radically changed, the student has the right to re-take the examination, which will be given twice during a one year period, according to the syllabus which was valid at the time of registration.

Other Information

The syllabus is a translation of a Swedish source text.