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Saturday, November 20, 2010

MST209 + MST326 as a 90 point option ?

It seems that MST209 is merely a ( level 2 ) introductory course on differential equations and that MST325 Mathematical Methods and fluid mechanics is the real deal on differential equations as far as the Open University is concerned.

Let me repeat the names of the courses once more:
MST209 - Mathematical methods and models
MST325 - Mathematical Methods and fluid mechanics
Just so that we agree that the word differential equations does not appear in either of the course names. - There is nothing wrong with changing a course description so that it will have appeal to a wider audience as long as the topic remains clear. I.e. "ODEs and PDEs" ( a name I can definitely imagine if the an occasional nerd slipped into the math community ) to "Ordinary and Partial Differential Equations", or: "Differential Equations" to "Mathematical modeling with Differential Equations". I do not understand the strange naming conventions of the Open University.

Names are just names lets have an in-depth look at the course descriptions.

MST209 - What you will study

This course will be of particular interest to you if you use mathematics or mathematical reasoning in your work and feel that you need a firmer grounding in it, or if you think you might find it useful to extend your application of mathematics to a wider range of problems. The course should also be suitable if you are teaching A-level applied mathematics, or if you intend to do so; the material on mechanics, in particular, gives a very careful treatment of the basic concepts of this subject. The teaching is supported and enhanced by the computer algebra package Mathcad.

Around half of this course is about using mathematical models to represent suitable aspects of the real world; the other half is about mathematical methods that are useful in working with such models. The work on models is devoted mainly to the study of classical mechanics, although non-mechanical models – such as those used in heat transfer and population dynamics – are also studied. The work on methods comprises topics chosen for their usefulness in dealing with the models; the main emphasis is on solving the problems arising in the real world, rather than on axiom systems or rigorous proofs. These methods include differential equations, linear algebra, advanced calculus and numerical methods. Many are implemented in Mathcad, so you can use the computer to solve more difficult problems and to investigate case studies.

The mechanics part of the course begins with statics, where there are forces but no motion, and then introduces the fundamental laws governing the motions of bodies acted on by forces – Newton's laws of motion. These are first applied to model the motion of a particle moving in a straight line under the influence of known forces. Undamped oscillations are discussed next. Newton's laws are then extended to the motion of a particle in space. The motions of systems of particles are modelled. Next we look at the damped and forced vibrations of a single particle. Then we look at the motion (and vibrations) of several particles. Finally, we investigate the motion of rigid bodies.

The methods part of the course covers both analytic and numerical methods. The analytical (as opposed to numerical) solution of first-order and of linear, constant-coefficient, second-order ordinary differential equations is discussed, followed by systems of linear and non-linear differential equations and an introduction to methods for solving partial differential equations. The topics in algebra are vector algebra, the theory of matrices and determinants, and eigenvalues and eigenvectors. We develop the elements of the calculus of functions of several variables, including vector calculus and multiple integrals, and make a start on the study of Fourier analysis. Finally, the study of numerical techniques covers the solution of systems of linear algebraic equations, methods for finding eigenvalues and eigenvectors of matrices, and methods for approximating the solution of differential equations.

MST326 - What you will study

In simple terms, we think of a fluid as a substance that flows. Familiar examples are air (a gas) and water (a liquid). All fluids are liquids or gases. The analysis of the forces in and motion of liquids and gases is called fluid mechanics. This course introduces the fundamentals of fluid mechanics and discusses the solutions of fluid-flow problems that are modelled by differential equations. The mathematical methods arise from (and are interpreted in) the context of fluid-flow problems, although they can also be applied in other areas such as electromagnetism and the mechanics of solids.

Because of its many applications, fluid mechanics is important for applied mathematicians, scientists and engineers. The flow of air over objects is of fundamental importance to the aerodynamicist in the design of aeroplanes and to the motor industry in the design of cars with drag-reducing profiles. The flow of fluids through pipes and channels is also important to engineers. Fluid mechanics is essential to the meteorologist in studying the complicated flow patterns in the atmosphere.

The course is arranged in 13 units within four blocks.

Block 1 is the foundation on which the rest of the course is built.

Unit 1 Properties of a fluid introduces the continuum model and many of the properties of a fluid, such as density, pressure and viscosity. The basic equation of fluid statics is formulated and used to find the pressure distribution in a liquid and to provide a model for the atmosphere.

Unit 2 Ordinary differential equations starts by showing how changes of variables (involving use of the Chain Rule) can be applied to solve certain non-constant-coefficient differential equations, and leads on to the topics of boundary-value and eigenvalue problems. It concludes with an introduction to the method of power-series for solving initial-value problems.

Unit 3 First-order partial differential equations extends the earlier version of the Chain Rule to cover a change of variables for functions of two variables, and shows how this leads to the method of characteristics for solving first-order partial differential equations.

Unit 4 Vector field theory relates line, surface and volume integrals through two important theorems – Gauss’ theorem and Stokes’ theorem – and formulates the equation of mass continuity for a fluid in motion.

Block 2 starts by investigating the motion of a fluid that is assumed to be incompressible (its volume cannot be reduced) and inviscid (there is no internal friction).

Unit 5 Kinematics of fluids introduces the equations of streamlines and pathlines, develops the concept of a stream function as a method of describing fluid flows, and formulates Euler’s equation of motion for an inviscid fluid.

Unit 6 Bernoulli’s equation analyses an important equation arising from integrals of Euler’s equation for the flow of an inviscid fluid. It relates pressure, speed and potential energy, and is presented in various forms. Bernoulli’s equation is used to investigate phenomena such as flows through pipes and apertures, through channels and over weirs.

Unit 7 Vorticity discusses two important mathematical tools for modelling fluid flow, the vorticity vector (describing local angular velocity) and circulation. The effects of viscosity on the flow of a real (viscous) fluid past an obstacle are described.

Unit 8 The flow of a viscous fluid establishes the Navier-Stokes equations of motion for a viscous fluid, and investigates some of their exact solutions and some of the simplifications that can be made by applying dimensional arguments.

Block 3 looks at a class of differential equations typified by the wave equation, the diffusion equation and Laplace’s equation, which arise frequently in fluid mechanics and in other branches of applied mathematics.

Unit 9 Second-order partial differential equations shows how a second-order partial differential equation can be classified as one of three standard types, and how to reduce an equation to its standard form. Some general solutions (including d’Alembert’s solution to the wave equation) are found.

Unit 10 Fourier series reviews and develops an important method of approximating a function. The early sections refer to trigonometric Fourier series, and it is shown how these series, together with separation of variables, can be used to represent the solutions of initial-boundary value problems involving the diffusion equation and the wave equation. Later sections generalise to the Fourier series that arise from Sturm-Liouville problems (eigenvalue problems with the differential equation put into a certain standard format), including Legendre series.

Unit 11 Laplace’s equation is a particular second-order partial differential equation that can be used to model the flow of an irrotational, inviscid fluid past a rigid boundary. Solutions to Laplace’s equation are found and interpreted in the context of fluid flow problems, for example, the flow of a fluid past a cylinder and past a sphere.

Block 4 returns to applications of the mathematics to fluid flows.

Unit 12 Water waves uses some of the theory developed in Block 3 to investigate various types of water wave, and discusses several practical examples of these waves.

Unit 13 Boundary layers and turbulence looks at the effects of turbulence (chaotic fluid flow) and at the nature of boundary layers within a flow, introducing models to describe these phenomena.

Now let's have a look at MIT 18.03, the introductory undergraduate course on differential equations at MIT:

18.03 - Description

This course is a study of Ordinary Differential Equations (ODE's), including modeling physical systems.

Topics include:
Solution of First-order ODE's by Analytical, Graphical and Numerical Methods;
Linear ODE's, Especially Second Order with Constant Coefficients;
Undetermined Coefficients and Variation of Parameters;
Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance;
Complex Numbers and Exponentials;
Fourier Series, Periodic Solutions;
Delta Functions, Convolution, and Laplace Transform Methods;
Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

I would say that, roughly, MST209 + MST326 = 18.03, but this means that I must add MST209 + MST326 to my options for 2011. I would not have seen this option if I hadn't been studying MST209 ( OpenLearn ) and the 18.03 lectures.


  1. MST326 is definitely a must if you want to extend your knowledge of partial differential equations. You know you can download the first unit and the revision booklet from the Maths and department statistics website. On the other hand the latter part of MST209 covers partial differential equations. M326 builds on this by including an introduction to the solution of differential equations by series. An introduction to special functions and the formulation of partial differential equations in other coordinate systems such as spherical polar and cylindrical. So you would need to be reasonably familiar with a lot of the material from MST209 before embarking on M326.

    What none of the course do as far as I can tell is solution of differential equations using Green's functions. This is basically where you turn a differential equation into an Integral equation. This technique is widely used in quantum mechanics especially quantum field theory.

    I plan to do MST326 as my last course for the OU maths degree. Starting in October 2012. Then embarking on the MSc in 2013 (so there will be a slight overlap).

    If you feel reasonably confident that you will have the background to do MST326 next year then go for it. I will definitely be interested to see how you find it.

    As for the naming conventions as MST209 also covers vector calculus and mechanics then its not just confined to ODE's or PDE's. So I think its a fair convention. Perhaps MST326 should be called something like Advanced mathematical methods and Fluid dynamics and MST209 Mathematical methods and mechanics. The other methods course is MS324 Waves, diffusion and Variation whioh covers the wave equation, the diffusion equation and calculus of variations.
    Again you can download the first block from the Maths and Statistics website.

    Best wishes Chris

  2. Thanks for the info Chris,

    I read about MS324 but it is not scheduled for 2011 and is scheduled for FINAL rollout in 2012.

    Do you mean by first block the Revision booklets?

    The revision booklet for MST325 gives a nice overview of what to expect from MST209 like vector calculus, functions of several variables and PDE's.

    I haven't decided yet about what I'll do next year but if it's going to be MST209 + MST325 it will be hard, especially in the beginning but what I see as a big plus is that it is going to feel as one (1) course, like MST121 / MS221. Info that I'll use as well in making a decision for 2011 is what I have been able to do on 18.03 around mid-December.

  3. As well as the revision booklet there is also the 1st 4 units of the course

    Should put you in a good position to make up your mind.

    If you look at the Library of Sample units

    Also the first block of MS324 on waves which I suspect will be retained when it gets merged with 325 to make 327.

    I don't think you would have much problem you seem to be on top of it all. Not sure what all this extra would do for your Pure maths ambitions though. You are correct they will merge together to feel like 1 course.

  4. Chris, Thanks for that link!

    You are right, but one can't deny Applied Mathematics. In any case: I'll be working ahead on M336 Groups and Geometry for 2012, so I have to 'fit that in' as well. ;-)

  5. I think the best type of mathematician is one who can combine the rigour of pure mathematics with the calculational facility of Applied maths. I know some pure mathematicians such as Hsrdy made great play of the fact that he was proud that his mathematics had no applications at all. However 15 years after he made that statement his theories were used in cryptography. So I think you are correct to combine a mixture of pure and applied maths courses. I guess you don't have to bother with Topology (unless you want to do functional analysis in the MSc) so that enables you to fit in some applied maths courses. M326 is an obvious one and I guess the Quantum mechanics course is also a good one to do.

  6. There are two movies in production about Ramanujan in which we will meet G.H. Hardy. I hope they have included dialogue where Hardy talks about pure vs applied math. One of the movies is based on "The Indian Clerk", which I might re-read, now that I think of it. I find reading biographical stuff of giants like Ramanujan, Turing, Riemann et al truly inspiring and it keeps my both feet on the ground if you know what I mean.

  7. Sounds interesting I'll look out for them. Theres are also good biographies of physicists such as Dirac, Heisenberg, Feynman, Gell Mann etc worth reading.


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