MECH4000 Advanced engineering mathematics Course description

This course provides a selection of advanced topics in Mathematics needed for later courses of the Master program. Topics include: linear algebra, with a focus on abstract inner product spaces, similarity transforms and diagonalization, as well as quadratic forms and positive definiteness. Vector calculus and applications to thermodynamics. Classification, and the solution of certain types of ODEs and PDEs using methods such as Laplace and Fourier transforms. Variational calculus with applications.

Students are expected to have prerequisite knowledge in mathematics and programming corresponding to national guidelines for a BSc degree in engineering.

Knowledge:   

The candidate

• can classify matrices in terms of positive definiteness and eigenstructure.
• can explain the relationship between properties of a quadratic form and its defining matrix.
• can explain how Euler’s formula dictates the behaviour of solutions to differential equations.
• can explain the variational operator in contradistinction to the differential operator and establish when the use of the former is required.
• can explain the Fourier series approximation in the context of projections onto function subspaces.
• can classify a PDE and outline methods to solve it including Fourier and Laplace transforms.

Skills:   

The candidate

• can calculate the eigenstructure of a square matrix and determine diagonalizability and positive definiteness.
• can apply the Gram-Schmidt process to obtain an orthormal basis for a given inner product space.
• can apply multiple integration and vector calculus to solve relevant engineering problems.
• can solve parabolic, hyperbolic, and elliptic PDEs using the methods of Fourier or Laplace transforms where applicable.
• can apply variational calculus to derive differential equations based on the minimization of the integral form.
• can use the principle of least action to derive the Euler-Lagrange equations of mechanics.

General competence:  

The candidate

• can communicate extensive, independent work, mastering the language and terminology of engineering mathematics.
• can discuss mechanical engineering problems with peers using the terminology of engineering mathematics.
• can contribute to innovative thinking and novel solutions precipitated by applying diverse mathematical modelling paradigms and classification methods.
• can model phenomena in solid mechanics, fluid mechanics and mechatronics using the language of mathematics, in particular PDEs, variational methods and linear transformations.

Lectures and tutorials (including problem solving and the use of Python coding to solving relevant mathematical problems).

None

Written 6-hour exam under supervision.

In the event of a resit or rescheduled exam, an oral examination may be used instead. If an oral exam is used, the examination results cannot be appealed.

A handheld calculator that cannot be used for wireless communication or to perform symbolic calculations. If the calculator’s internal memory can store data, the memory must be deleted before the exam. Random checks may be carried out.

Graded scale A-F

One internal examiner. External examiner is used periodically.

Tore Flåtten