Mathematics

Module aims

  • To introduce the basic concepts of variational calculus and their applications
  • To introduce the theory of complex variables and techniques based on complex variables

Learning outcomes

Knowledge and understanding
On successfully completing this course unit, students will be able to:

  • Formulate and solve the Euler-Lagrange equations;
  • Apply the basic theory of functions of a complex variable;
  • Derive Cauchy’s theorem and evaluate the integral of a simple complex function around a curve in the complex plane;
  • Derive power series expansions of complex functions about singular points of the functions;
  • Distinguish between the different types of singularities that can arise in the complex plane;
  • Derive Cauchy’s residue theorem and use it to evaluate real integrals over finite and infinite ranges;
  • Calculate inverse Laplace transforms using the residue theorem and apply this technique to solve certain partial differential equations;
  • Use the ideas of conformal mappings to solve equations in geometrically complicated domains (e.g. to determine the inviscid flow field around a Joukowski airfoil).

Skills and other attributes
On successfully completing this course unit, students will have acquired the following skills:

Intellectual skills

  • The ability to think clearly and pay attention to detail;
  • The ability to manipulate expressions algebraically, and minimize the making of errors.

Practical skills

  • Problem-solving: the ability to formulate a problem precisely and then solve it logically, making all assumptions clear.
  • Investigative skills: e.g. researching material on-line and in the library, asking others for advice.

Transferable skills

  • Communication skills – to be successful you will need to develop your listening and note-taking skills.
  • Determination – to solve the most difficult problems you will need to persevere.
  • Creativity – some problems require a combination of techniques.
  • Intellectual rigour.
  • Ability to work independently.

Module syllabus

Calculus of variations:  Euler-Lagrange equations; problems with constraints.
Functions of a complex variable:  Revision of complex numbers: triangle inequality, polar coordinate representation, curves in the complex plane. Continuity and differentiability of complex functions: analyticity, the Cauchy-Riemann equations. Definitions and properties of elementary complex functions. Branches and branch points. Complex line integrals: definition and properties. Cauchy’s integral theorem and its consequences. Cauchy’s integral formula. Complex power series: Taylor series, Laurent series. Classification of singularities in the complex plane: poles, residues and essential singularities. The residue theorem: contour integration, evaluation of real integrals.
Laplace transforms: Revision of basic properties. Derivation of complex inversion formula. Use of contour integration. Application to differential equations.
Conformal Mapping: Application to Laplace’s equation. The Joukowski transformation.

Pre-requisites

AERO40006 Mathematics 1 
AERO50006 Mathematics 2

Teaching methods

Lectures, tutorials. Teaching via printed material, whiteboard and interactive online animations/demonstrations.

Assessments

Examined Assessment
2 hour written examination in the Summer term  (100%)

Non-Examined Assessment
Marked coursework.
 

Reading list