Mathematics for engineering 2 A

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.

Laplace transform. Computation by hand and computer. Applications of the Laplace transform to differential equations and signal processing.

Programme module. Trigonometric series and Fourier series. Applications to 1D wave equation with separation of variables. Fourier transform. Computation by hand and computer. Applications of Fourier transforms. Spectral analysis (e.g. sound and light waves). Applications to solution of differential equations, including harmonic motion with external periodic forcing.

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.
• Laplace transforms and applications to differential equations and signal processing.
• Series representations and approximations to functions, particularly Taylor and Fourier series.
• Fourier transforms and applications to spectral analysis
• 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
• Laplace transform certain functions with applications to solution of differential equations and signal processing
• Compute Fourier coefficients of functions
• Fourier transform certain functions with applications to solution of differential equations and spectral analysis
• 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.

Lectures, exercises and a project.

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