MST209 Kick Off

Finished and approved the plan for MST209 today. Status: in-progress.

I have been revising some physics: classical ( Newtonian ) mechanics. Amazing what we can do with three simple laws and a few calculus tools: the foundation of modern technology.

MST209 - Overview Blocks Units

I'll be doing MST209 next year. It is the most logical choice after MST121, MS221 and M208 the courses I have done sofar. The course looks quite interesting if you ask me. I already studied some topics based on the OpenLearn version of MST209 supported by corresponding MIT 18.02 / 18.03 video lectures. Although I don't study mathematics because of the stuff offered in MST209 I accept that MST209 is part of the mathematical language shared by all scientists ( and engineers ). Besides that MST209 is packed with examples I can play with in Mathematica, do some Mathematica programming as well.

Since I don't have the course materials yet ( haven't even registered yet ) I made a simple planning in a spreadsheet. Blocks / Units as rows, weeks as columns. The TMA cut-offs are estimated based on M208 data of this year. My study 'capacity' in net study hours is roughly as follows on a weekly basis.

NET STUDY(=RESOURCE) CAPACITY

Mon	Tue	Wed	Thu	Fri	Sat	Sun	Total
2	6	2	2	-	6	6	= 24

Total 24hrs or 90P / year. ( Read: Work + Study and do nothing else. )

I scheduled the MST209 data in the schedule and it fits very well. I usually work two weeks on a TMA. Because I use LaTeX, do a lot of quality-checks, and start asap on a TMA, which means I might need to study stuff while working on the TMA. Anyway this means that I have two weeks to study a block, 5 Units or roughly 20 sections, do exercises, etc. You all know the drill.

Working on a degree is basically an excuse for spending time on math. In most circles it is considered anything between nerdy, nutty and plain vanilla crazy. ( Ignorance is NOT bliss. ) I am just explaining that my plan objective is not completing the project at the earliest possible date.

I have the following options.
- Blank MST209
- MST209 + M381 Number theory and logic ( As in my original overall plan )
- MST209 + M337 Complex analysis
- MST209 + MST326 Mathematical methods and fluid mechanics

My next step is, based on my study and OU experiences sofar, trying to 'fit in' another 30 point course ( any course for that matter ). If ( and only if ! ) that works I have to choose which module I will do next to MST209.

What's in MST209 :

BLOCK-1
Unit 1 Getting Started
This unit focuses mainly on mathematical techniques, but also covers some examples involving skills in the application of mathematics.

Unit 2 First-order Differential Equations
This unit considers in detail how a differential equation arises in a mathematical model with basic definitions and terminology associated with differential equations and their solutions.

Unit 3 Second-order Differential Equations
This unit considers second-order differential equations, that is, differential equations that involve a second (but no higher) derivative.

Unit 4 Vector Algebra
We often need to represent physical quantities such as mass, force, velocity, acceleration, time, etc., mathematically. Most of the physical quantities that we need can be classified into two types: scalars and vectors. This unit defines a vector and discusses ways of representing vectors in two and three (or more) dimensions. Also considered are ways of operating on and combining vectors - that is, they provide the fundamentals of vector algebra.


BLOCK-2
Unit 5 Statics
This unit and unit 6 lay the foundations of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. This unit considers how and why they move. This unit assumes a good working knowledge of vectors.

Unit 6 Dynamics
Continuing on from Unit 5 this unit considers how and why objects move and outlines the procedure for solving dynamics problems. There is a video sequence associated with section 2 of this unit and is available on the DVD (order code MST209/DVDR01).

Unit 7 Oscillations
All around you there are mechanical systems that vibrate or oscillate. Each day you probable experience oscillations in a wide variety of forms: the buzzing of an alarm clock, the vibrations of an electric hair-drier or razor, the sideways movements of a train or boat, and so on. This unit describes a simple experiment involving an oscillating system, introduces Hooke's law as a model for the force exerted by a spring, and goes on to consider how this law applies in various situations where no movement takes place. It also shows how to use Newton's second law to model the oscillations of the simplest oscillating system, which consists of a single particle attached to a single horizontal spring.

Unit 8 Energy and Consolidation
This unit consolidates the mechanics covered in the previous units and introduces the topic of energy in mechanical systems.


BLOCK-3
Unit 9 Matrices and Determinants
This unit examines some of the properties and applications of matrices. It introduces the matrix method of solving large systems of linear equations, called the Gaussian elimination method, and explains the conditions required for this method to work.

Unit 10 Eigenvalues and Eigenvectors
This unit introduces eigenvectors showing simplified problems. It considers the eigenvectors and eigenvalues associated with various linear transformations of the plane and outlines situations where eigenvectors and eigenvalues are useful.

Unit 11 Systems of Differential Equations
This unit focuses on systems of linear differential equations relating two more functions and their derivatives. It shows how various situations can be modelled by a system of linear differential equations, how such a system can be written in matrix form, and how eigenvalues and eigenvectors can be used to solve it when the equations are homogeneous with constant coefficients.

Unit 12 Functions of Several Variables
This unit extends the calculus of functions of one variable to functions of several variables. It also discusses the application of functions of several variables to mechanics.


BLOCK-4
Unit 13 Modelling with Non-linear Differential Equations
In this unit we study the mathematical models associated with two physical systems: the growth of two interacting populations, one a predator and the other its prey and the motion of a rigid pendulum.
This is demonstrated through the use of Lotka-Volterra equations, which apply to a pair of interacting populations. How these equations can be linearized near an equilibrium state and the graphical representation of the solutions are discussed.

Unit 14 Modelling Motion in Two and Three Dimensions
This unit turns the attention to motion and forces in more than one dimension. It draws on ideas about vectors, forces and component forces and fundamental ideas about mechanics of particles, in particular Newton's second law. There is also mention of kinetic energy and potential energy. The video sequences associated with this unit are available on the DVD (order code MST/209/DVDR01), however it is not essential for you to view these.

Unit 15 Modelling Heat Transfer
This unit makes use of ideas relating to energy and first-order differential equations. It begins by developing models that could be used to answer questions such as the following.

How much does it cost to heat up a tank full of hot water?
What thickness of insulation should be applied to a hot-water tank?
What thickness of insulation should I use in my loft, and what savings would I make over a year?
What should the gap be in double-glazing?

Is it better to insulate the roof, insulate the walls or double-glaze the windows of my house?
The common factor in answering all these questions is the need to consider the transfer of heat energy between different regions of space. This unit introduces the basic ideas of heat energy and temperature. It discusses conduction and convection as well as a third mode of heat energy transfer, radiation.

Unit 16 Interpretation of Mathematical Models
This unit introduces the ideas of mathematical modelling. It discusses the five key stages of the mathematical modelling process in detail and looks at dimensions and units of physical quantities to see how they can be used to predict and check the outcomes of the modelling process.


BLOCK-5
Unit 17 Damping, Forcing and Resonance
This unit refers back to the contents of several earlier units. In particular it builds upon and extends the model of simple harmonic motion and uses this approach in analysing one-dimensional motion. It also returns to the concept of a resistance force proportional to the velocity of a particle.

Unit 18 Normal Modes
This unit continues with the theme of mechanics, in particular, it builds on earlier units that dealt with oscillations. In order to solve the equations of motion derived for the mechanical systems studied, it uses the methods used for solving systems of differential equations. This unit also draws heavily on the discussions regarding eigenvalues and eigenvectors.

Unit 19 Systems of Particles
The main objective of this unit is to show how to obtain useful information on the complicated motion of an object or system, and demonstrates that the concept of centre of mass is crucial to this process.

Unit 20 Circular Motion
The theme of this unit is rotational motion. It concentrates mainly on analysing the circular motion of a particle. This can be used to model a wide range of situations, such as a child on a swing, the pendulum of a clock and a chair-o-plane roundabout at a fairground. This unit builds on many of the ideas from earlier units mainly: polar coordinates, vectors, torque and Newton's second law.


BLOCK-6
Unit 21 Fourier Series
This unit is concerned with the technique of expressing a periodic function as a sum of terms, where each term is a constant, a sine function or a cosine function. This unit assumes you have a background knowledge of the definition of the period (unit 7), forced oscillations and resonance (Unit 17, and integration by parts (Unit 1).

Unit 22 Partial Differential Equations
This unit builds on ideas previously introduced in unit 12 regarding The diffusion equation and the wave equation, in the context of modelling the vibrations of a taut string (such as guitar string).

Unit 23 Scalar and Vector Fields
The main focus of this unit is the differential calculus of scalar and vector fields, i.e. the study of how scalar and vector fields vary from one point to another. A brief introduction to the properties of orthogonal matrices and pictorial representations of scalar and vector fields is given along with an extended discussion of the gradient function of a scalar field. Cylindrical and spherical polar coordinate systems for specifying points in three dimensions is also introduced in this unit.

Unit 24 Vector Calculus
This unit discusses the divergence of a vector field, the curl of a vector field, the scalar line integral and linking line integrals curl and gradient. This unit also builds on the concepts of kinetic energy and potential energy.


BLOCK-7
Unit 25 Multiple Integrals
This unit generalises the idea of an integral still further to deal with two and three dimensions by introducing two new kinds of integrals, called area integrals and volume integrals. How area integrals can be evaluated as combinations of two ordinary integrals, is shown and applications of area integrals, including the evaluation of centres of mass of planar (i.e. two-dimensional) objects are described. How volume integrals can be expressed as combinations of three ordinary integrals and how area integrals can be used to compute the area of a curved surface are also demonstrated.

Unit 26 Numerical methods of Differential Equations
This unit introduces the study of numerical methods for differential equations. It covers the Taylor's theorem with exercises, recaps Euler's method for solving initial-value problems involving first-order differential equations and goes on to explain that more efficient methods exist. Three new methods known as Runge-Kutta methods, are derived and a way of determining how small the step size h would need to be in order to achieve a given accuracy for a given initial-value problem is established.

Unit 27 Rotating Bodies and Angular Momentum
This unit deals with the motion of extended bodies, and in particular with their rotational motion. Rotating bodies, Angular momentum, Rigid-body rotation about a fixed axis and rotation about a moving axis are all covered in this unit.

Unit 28 Planetary Orbits
This unit shows how Newton's laws of motion and Newton's law of universal gravitation can be used to predict the orbits of planets around the Sun. In particular, it shows that Kepler's laws of planetary motion can be derived using Newtonian mechanics.

Watched MIT 18.02 - lecture 11

In 18.02 lecture 11 Prof. Denis Aroux talks about differentials and the Chain Rule. Two of the examples used to illustrate the main topic are of particularly interesting: a new proof for the differentiation of products and quotients, and the conversion between rectangular and polar coordinates.

A main result of this lecture is $$df = f_u \frac{du}{dx} + f_v \frac{dv}{dx},$$ where $f$ is a function of two variables $u,v$ which are both dependent on $x$, and $f_u, f_v$ are partial derivatives. The quotient rule can be derived from this result as follows. Let $g(x) = \frac{u}{v}$, with $u,v$ both dependent on $x$ :
$\begin{align*}

df &=f_u \frac{du}{dx}+f_v \frac{dv}{dx} \\
&=  \frac{1}{v}\frac{du}{dx}-\frac{u}{v^2}\frac{dv}{dx} \\
&= \frac{ v \frac{du}{dx}-u \frac{dv}{dx} }{v^2}
\end{align*} $
The last expression is the quotient rule for differentiation.

This lecture inspired me to some experimentation ( play ) with Mathematica's PolarPlot function. A polar coordinate is in fact a function of two variables $x,y$ which are both dependent on $r$ and $\theta$ with $x=r \cos(\theta)$, $y=r \sin(\theta)$. By applying the theory above one suddenly gets control over geometric objects like this:

Click to enlarge

Finally the concept of a gradient was mentioned which is merely a vector of partial derivatives. Gradients are the topic of lecture 12. I designed some problems and exercises ( and other experiments ) for functions in polar coordinates. I am delighted I feel more able in that regard.

Watched MIT 18.02 - lectures 9,10

The Second Derivative Test is a procedure for determining if a critical point is a maximum, a minimum, a saddle or a degenerate.

PROCEDURE:
- Calculate $f_x, f_y, f_{xx}, f_{xy}, f_{yy}$
- Calculate the critical points ;

Then for each critical point:
- Calculate $A = f_{xx}(x_0, y_0)$
- Calculate $B = f_{xy}(x_0, y_0)$
- Calculate $C = f_{yy}(x_0, y_0)$
- $AC-B^2$.

Apply the Second Derivative Test
If $AC-B^2 > 0$ and $A > 0$ minimum
If $AC-B^2 > 0$ and $A < 0$ maximum
If $AC-B^2 < 0$ saddle
If $AC-B^2 = 0$ degenerate

Click to enlarge

EXAMPLE:
( See Mathematica print )
$f(x,y)=e^{x^2-\frac{x^4}{4}-y^2}$

At $(0,0): \left\{AC-B^2, A\right\} = \left\{-4,2\right\}$
Saddle.

At $(-\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

At $(\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

.

Watched MIT 18.02 - Lecture 8

A lecture in which Prof. Denis Auroux says: "Applied mathematics is physics with a different set of symbols." To the math.

Lecture 8 starts a new unit in the 18.02 series and is about functions of several variables, the graph of a function of two variables, contour plots of functions of two variables and partial derivatives.

The image above shows the graph and the contour plot of $f(x,y)=1-(x^2+y^2)$ followed by the partial derivatives as calculated by Mathematica.

Partial derivatives are defined as follows:
$$\frac{\partial f}{\partial x}(x,y) = \displaystyle\lim_{\Delta x \to 0}\frac{f(x+\Delta x, y) - f(x,y)}{\Delta x}$$
$$\frac{\partial f}{\partial y}(x,y) = \displaystyle\lim_{\Delta y \to 0}\frac{f(x, y+\Delta y) - f(x,y)}{\Delta y}$$

.

[TIP] - Join a Study Group

I joined an MIT 18.02 Online Study Group since I think 18.02 and MST209 have many similar topics. There are several other Study Groups as well. If you have a Facebook account you can login with your Facebook Id.

http://openstudy.com/

MIT 18.02 Study Group

More on MST209 in relation to MIT video lectures

MIT has a video lecture series on multivariable calculus: 18.02, which is in fact a prerequisite for 18.03. I looked further into the topics of MST209 and I now think that 18.02 is a much better preparation for MST209 than 18.03.

It's more or less like this:
MST209 = 18.02 + ( part of ) 18.03
MST209 + MST326 = 18.02 + 18.03

18.02 has lectures on
Lecture 15: Partial Differential Equations
Lecture 16: Double Integrals
Lecture 19: Vector Fields
Lecture 21: Gradient Fields
Lecture 25: Triple Integrals
Lecture 27: Vector Fields in 3D
Lecture 30: Line Integrals
which are topics in MST209.

18.02 Multivariable Calculus

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.

Difference 18.03 - MST209 ( OpenLearn edition )

Taken from MST209 (OpenLearn edition) Book 2: Second order differential equations:





The 18.03 method however is as follows:

$2\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=2e^{-x}$

$(2D^2-2D+1)y=2e^{-x}$,
( where $\alpha=-1$ and $p(D)=2D^2-2D+1$ )

$y_p = \frac{2}{p(\alpha)}e^{-x} = \frac{2}{5}e^{-x}$

I don't know if the linear differential operator ($D$) is discussed in any part of MST209, if not the conclusion is simple: MIT 18.03 does a better job at teaching differential equations than MST209 since Matuck said that he $D$ operator plays an important role in the rest of 18.03 ( lectures 14-33 ).

Watched MIT 18.03 - lecture 9

I skipped lectures 7,8 (watched 7 partially and had a brief look at 8 ) because during this first serious confrontation with differential equations I want to follow the route of MST209 ( OpenLearn version ).

In 18.03 lecture 9 Prof. Mattuck talks about differential equations of type $y'' + Ay' + By = 0$. They are of the 2nd order, have constant coefficients and are homogeneous. The procedure for solving them is surprisingly similar to solving second order recurrence equations. The DE has a characteristic equation $r^2 + Ar + B=0$. If both roots are real and distinct the general solution then looks like $y=c_1 \cdot e^{r_1x} + c_2 \cdot e^{r_2x}$. The other two cases ( i.e. a pair of complex conjugates, two real equal roots ) are discussed during the rest of the lecture.

Change of plan

Since I am still in the planning phase I can change the plan whenever I think the overall sitiatopm nemefirs

I changed my mind on the books I will use next to the MST209 materials. If I don't understand something in book A I have a look how it is explained in book B. Comparing books also helps in determining what's really important. I am not ready yet for 'Kelley, Peterson' so I am going to drop their book.

- Elementary Differential Equations Sixth Edition C. Henry Edwards David E. Penney ( Used in MIT 18.03 )
- J. David Logan A First Course in Differential Equations ( Springer book, formal at the undergraduate level )
- Differential Equations with Mathematica Third Edition Martha L. Abell James P. Braselton ( Very practical with tons of do-able exercises in Mathematica )
- Differential Equations Demystified Steven G. Krantz ( Lots of Recipes, Exercises and Solutions )

All in all a balanced set of books to complement the core MST209 materials, I suppose.

Study plan for 2011 is shaping up

The Study Plan for 2011 is shaping up.

With M336 Group Theory forthcoming in 2012 I will keep my Abstract Algebra 'warm' by reading:
- Goodman; Algebra Abstract and Concrete. ( Free e-book );
- Rose, Harvey E; A Course on Finite Groups;
- Cox, David; Galois Theory;

I will probably have some difficulty ( 'uneasyness', if you like ) studying MST209 because the materials will not be presented in mathematical but in scientific format. Since I am not very good at long reading sessions ( I get distracted too easily ) I prefer the Theorem / Proof / Example presentation because it leaves a lot to the reader, i.e.: read a bit, then DO a lot. Mathematica experiments for example. For that purpose I selected the books
- Kelley, Peterson; The Theory of Differential Equations - Classical and Qualitative. From the preface: "...Previous experience with differential equations is helpful but not required. Consequently, this book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students. ...)", and
- Differential Equations DeMystified as an antidote to the previous book and as a guide to the HOW-TO's of solving DE's.

Top priority will of course have MST209 because it adds 60 points to my balance. I'll have to decide if I want and / or can add either M337 Complex Analysis or M381 Number Theory to the workload.

Analytical function representing the Fibonacci series

The MST209 exam has a different format than MS221 and M208 have. In 2006, for example, the format was as follows:
Part A. 15 multiple-choice questions, 2 marks each = 30 points. ( 1 hour )
Part B. 8 questions, 5 marks each = 40 points. ( 1 hour 15 min )
Part C. 3 out of 7, 15 marks each = 45 points. ( 45 min )
Yes. Maximum score is 115. Scores above 100 are set to 100.
The exam looks doable. And again questions on eigenvalues and eigenvectors. That would be three in a row: MS221, M208 and MST209. Considering the fact that one can prove Binet's formula for the Fibonacci numbers with them it's worthwhile having it firm under your math-belt.

There is also an analytical function for the Fibonacci numbers which rounded, gives an exact Fibonacci number if the input variable is an integer. Here is the related math.

Let $F_n = F_{n-1} + F_{n-2}, F_0=0, F_1=1$, show that $Fa_n=\frac{1}{\sqrt{5}}e^{n \cdot \log{\phi}}$, where $\phi$ is the Golden Ratio or $\frac{1+\sqrt{5}}{2}$. ( Round $Fa_n$ to get $F_n$. )

If we define the elements $F_{n}$ and $F_{n+1}$ as the vector $s_n= \left(
\begin{array}{c}
F_{n+1}\\
F_{n}
\end{array}
\right)$
then $F_n$ simply becomes
$F_n= \left(
\begin{array}{cc}
1 & 1\\
1 & 0
\end{array}
\right)^n
\cdot s_{0}$.

We can calculate the power of a matrix by diagonalizing the matrix. And this is where eigenvalues and vectors come in. If $\lambda_1, \lambda_2$ are eigenvectors with respective eigenvectors $E= \left( e_1, e_2 \right)$ we get $F_n= E^{-1}
\cdot
\left(
\begin{array}{cc}
\lambda_1^n & 0\\
0 & \lambda_2^n
\end{array}
\right)
\cdot
E
\cdot s_{0}$

The eigenvectors are the roots of the characteristic equation $\left|
\begin{array}{cc}
1-\lambda & 1\\
1 & -\lambda
\end{array} \right| = 0$ and are thus $\frac{1}{2} + \frac{1+\sqrt{5}}{2}$ and $\frac{1}{2} - \frac{1+\sqrt{5}}{2}$.

( TO BE CONTINUED ... )

Watched MIT 18.03 videos 2 and 3.

Video 3 is about a straightforward recipe for solving a linear ODE of the first order:


Video 2 is on numerically solving DE's with Euler's method:

Getting started on MST209 Differential Equations

In order to get a feeling for MST209 I will be watching the video's of MIT 18.03 Differential Equations by Prof. Arthur Mattuck. I watched lecture 1 today. Since there are 33 video lectures in total I doubt if I'll get through all of them by february though. The purpose is getting back into differential equation stuff. OU OpenLearn has several booklets from the MST209 course available which give a reasonable idea on what to expect of MST209. Both MIT 18.03 and MST209 are less formal than M208, they are mainstream courses teaching mainly how-to's, i.e. how-to solve a particular type of differential equation. - My choice for MST209 next year is more or less certain by now. Three options remain. 1) do just MST209 and concentrate and spend time on self-study of topics that really interest me (i.e. Abstract Algebra stuff ). 2) Add M381 Number Theory + Logic. 3) Add M337 Complex Analysis. Well, I haven't decided yet. MST209 involves 7 TMA's, 2CMA's and probably a MS221, M208 'type-of-exam', i.e. 12 questions for 70, plus 2 out of 5 for 30.