Course content

Basis module. Functions of several variables. Partial differentiation, gradient. Critical points and optimization. Taylor’s theorem with remainder. Introduction to partial differential equations: examples and solutions.

Partial differential equations. Different types required different approaches, focus on physical/modeling intuition. Overview of the field. Steady state equations. Examples: Laplace’s and Poisson’s equation. Solution by computer using linear algebra. Iterative numerical solutions converging to a steady state. Time-dependent systems. Examples: Heat equation, advection equation, wave equation. Solution by computer.

Programme module. Optimization without constraints. Methods utilizing the derivative. Method of least squares - linear and non-linear. Optimization with constraints. Lagrange multipliers. Linear programming. The dual problem. Solutions by simplex method on computer. Integer programming.

Learning outcome

Knowledge

The candidate has good knowledge of:

• Functions of several variables, including partial derivatives and their application to classification of stationary points and optimization.
• Taylor’s theorem and approximation by Taylor series.
• Partial differential equations, their properties and applications.
• The most important concepts and methods from optimization, such as iterative methods, constraints, Lagrange multipliers, objective function, dual problem.
• Digital tools for analysis of mathematical problems.

Abilities

The candidate can:

• Find and interpret the partial derivatives of a function of several variables
• Approximate functions by Taylor’s theorem and estimate the error with a remainder term.
• Solve simple optimization problems with several variables.
• Verify that a given function solves a partial differential equation
• Solve certain partial differential equations by computer, verify and interpret the results.
• Use computers for optimization without constraints and interpret the results.
• Solve simple optimization problems with constraints using Lagrange multipliers.
• Formulate applied problems as linear programming and solve by computer and interpret the results.
• Apply digital tools to analyse mathematical problems.

General competence

The candidate:

• Has good knowledge of, and can apply a symbolic and formulaic mathematical apparatus that is relevant for communication in engineering sciences
• Has experience of evaluation of their own and other students scientific work, and with giving precise and technically correct oral feedback.
• Has experience with applications of mathematical methods and digital tools to problems with their own and related specializations.
• Can connect mathematical concepts and techniques to models the candidate meets within and outside of their studies.

Learning methods and activities

Lectures, exercises and group work.

Compulsory assignments

The compulsory assignments consist of two parts:

• Compulsory exercises that are based on both analytical and numerical solution of problems and interpretation of the results. The assignments include tasks to be solved with the help of digital tools.
• Compulsory group work

Special conditions

Obligatory activities from previous semesters can be accepted by the institute.

Compulsory assignments

• Compulsory assignments (exercises and group work)

Specific conditions

Recommended previous knowledge

Mathematics for engineering 1 or similar

Course materials

Recommended course material will be announced at the start of the semester.

Credit reductions