F19AB2 Applied mathematics B

(Third year module given in the second semester.)

Introduction

Lecturer. Simon Malham, Room CM T.21, Mathematics Department.

Contact. email: S.J.A.Malham [at] hw.ac.uk and tel: 0131 451 3254.

Lectures. Monday 1:15pm in EM183, Thursday 9:15am in EM182 and Friday 11:15am in EM183.

Tutorials. Thursday at 3:15pm in CMB S01.

Vision. You can find all the lecture notes, handouts, solutions to exercises and past papers and so forth on the course VISION page.

Course aims and objectives. The objective of the module is to introduce some fundamental ideas and techniques in Applied Mathematics.

Syllabus.

• Calculus of variations: Calculus of variations: variational derivative; Euler-Lagrange equations; examples including the Brachistochrone, isoperimetrical, and soap bubble problems; extensions to higher derivatives, several dependent variables; constraints and Lagrange multipliers. (6 lectures)
• Lagrangian mechanics: Action; Hamilton's Principle; Lagrange's equations; examples including the Kepler and simple pendulum problems; derive incompressible Euler equations; Hamilton's equations; Poisson brackets and the Hamilton-Jacobi partial differential equation. (5 lectures)
• Fluid equations: Continuum Hypothesis; Lagrangian and Eulerian formulations; material derivative; continuity equation; balance of momentum; Transport Theorem; Equation of State; incompressibility. (4 lectures)
• Flows and applications: isentropic fluids and Bernoulli's Theorem; streamlines; vorticity; Couette flow. (3 lectures)
• Fourier Analysis: Full and half range Fourier series (3 lectures)
• An introduction to PDEs: Simple PDEs; Separation of Variables; Solution of Heat equation, Laplace's equation and the wave equation making use of Fourier series. (8 lectures)

Assessment. The continuous assessment consists of a one hour midterm exam counting for 10 percent of the final mark, homework counting for 5 percent of the final mark (details below) and a two-hour final exam at the end of term which counts for 85 percent of the course mark. The midterm will be held on

Thursday February 18th

The homework assessment consists of the specified exercises in the table below, to be handed in before or on the dates indicated. You can score up to 20 marks per homework. Your best 5 homeworks out of the 7 will be added together to generate your overall continuous assessment score (for maximum credit you need to score a total of 100 marks).

There is a resit in August for the ordinary course. The resit assessment is purely on the basis of a two-hour exam.

Contract. Students are expected to read the notes before, during and after the lectures and tutorials. Lectures will act as a more formal forum for the lecturer to explain the ideas of the course and give alternative examples, whilst tutorials will take a less formal and more personal form. There are exercises at the end of each chapter and students must attempt these. Mathematics is best learned through grappling with the underlying ideas presented in lectures and then tackling problems given in the exercises.

You cannot learn to swim by reading a book about it!

Hence try the exercises, and if you get stuck, ask the lecturer either after a lecture, during the tutorials. It is vital that you can solve problems proficiently. If you need help, then

Attendance sheets. Students will be required to sign an attendance sheet with their initials in every lecture and tutorial. If any one student misses three consecutive such contact events, or more than one-third of them overall up until that date, then their personal mentor will be contacted.

Evaluations. At the end the course students will have an opportunity to fill out formal university evaluations on the course.

Books. The two main recommended books are V.I. Arnold and Chorin and Marsden (see the bibliographies of the lecture notes for details).

Announcements

• The midterm exam will be in week 6 on Friday February 19th at 11:15am in EM183. It lasts for one hour. It will cover the lectures on Chapter 1: Calculus of variations and Chapter 2: Lagrangian and Hamiltonian mechanics.

Electronic resources

Syllabus (from the official department course pages)

Movies. These are the movies shown during the course. Download them and use them freely.

Homework timetable

This may change slightly as we progress so keep checking this webpage.

There are 7 homeworks here, each is worth 20 marks. Your best 5 will be used to make your final score (which is worth 5% of your overall mark for the module).

 Exercises Date due Euler-Lagrange alternative form + Soap film Jan 29th Hanging rope Feb 5th Central force field Feb 12th None---midterm this week. Feb 19th 1.4 Steady Oscillating Channel Flow and 1.5 Channel Shear Flow Feb 26th Bernoulli Theorem execise to be handed out in class Mar 11th Fourier series: question 1 Mar 18th Heat equation: question 1 Mar 25th

This webpage and its content was started on 30/1/2009.

Please feel free to download and use any of the material accessible from this page---provided that it is not used for commercial gain.

Last updated: 18/2/2016.

S.J.A.Malham [at] hw.ac.uk