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. 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 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 a project.

Tasks require both analytical and numerical methods with the use of digital tools.

Compulsory assignments

• Compulsory assignments (exercises and a project)

Specific conditions

Recommended previous knowledge

Mathematics for engineering 1 or similar.

An introductory course in Python.

Course materials

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

Credit reductions