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Introduction to Physical Modeling (MATH 11008)

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Contents of this document:

Administrative Information

  1. Unit number and title: MATH 11008 Introduction to Physical Modeling
  2. Level: C/4 (Honours)
  3. Credit point value: 10 credit points
  4. Year: 07/08
  5. First Given in this form: 2007/2008
  6. Unit Organiser: Lorena A. Barba
  7. Lecturer: Dr L Barba
  8. Teaching block: 1
  9. Prerequisites: An A in A-level Mathematics, or equivalent. Calculus is a co-requisite.

Unit aims

This course will give you a sense of the importance of mathematics in the world, and will show you how an applied mathematician can help answer some big question that humanity needs answered.

 

General Description of the Unit

When applying mathematics to the “real world”, often we deal with situations in which some quantity is changing in time. These situations are dynamic, and we study in applied mathematics “dynamical systems”. If we need to know what yearly payments a country needs to make to pay off its loan to the international monetary fund within a generation, or if we would like to estimate the rate at which the human body eliminates a drug compound, we will use this type of mathematics. In this unit, we will open a window to this world of using mathematics to investigate the systems of biology, economics, pharmacology, genetics, etc.

 

Relation to Other Units

Some of the material in this unit will be revisited, but studied at greater depth, in the Ordinary Differential Equations unit. Interested students may want to gain deeper understanding of some of the applications, for example, attending the units Financial Mathematics, or Nonlinear Dynamics and Chaos.

 

Teaching Methods

Lectures - 22 sessions in which the lecturer will present the course material via computer using tablet technology. Sometimes, computer demonstrations or short presentations will be included. Handouts will be given and discussed. Students are expected to attend all lectures, and to prepare for them by reading notes, handouts or texts, as indicated by the lecturer and/or announced in the course Blackboard space (see below). The lectures are 2 per week, on weeks 1 to 11 - no class on week 12 .


Problems classes - fortnightly sessions with the lecturer, in which problems will be worked through as a demonstration, on the projected computer screen, using tablet technology. Students are strongly encouraged to attend all problems classes.


Computer lab - sessions conducted by the student teaching assistants. The class will be divided in several groups, which will meet in different sessions at the Undergraduate Computer laboratory. Students will perform tasks related to the homework problems and lecture examples. Students are strongly encouraged to attend all lab sessions. The computer lab sessions will prepare the students for their course project work (see below). In addition to the work in the lab, students will require further personal work using computers, as part of the homework problems.

Homework assignments - 10 problem sheets will be given out, one per week. Students will be required to turn in selected problems from the sheet, which will be marked by the postgraduate teaching assistants. The best 8 of 10 problem sheets will contribute 30% to the final mark for the course . No excuses for late turn-in will be accepted (if a homework set is not turned in due to sickness, that homework will be one of the 2 which are discounted for the grade).


Blackboard - the course will use the University's Blackboard environment to distribute material (handouts, reading assignments, problem sets, solutions, etc.), and for announcements. The environment also provides for a Discussion forum for students to carry out conversations online.


Course Project - the course project consists of an investigation using computers. The mark obtained on the project report will count 10% of the course. Students are allowed, and even encouraged, to work in pairs for the course project, and to turn in one report for each two-people team. Additional instructions will be available in the Blackboard space in due course.

Learning Objectives

By the end of this course, you will:

  1. Have an appreciation for how mathematics is used in the “real world”.
  2. Understand the concepts of a dynamical system, a model, dependent and independent variable
  3. Grasp the concepts of stability, oscillations, equilibrium
  4. Understand the difference between constant and proportional change.
  5. Be able to make the connection between constant change and linear function, between proportional change and exponential function.
  6. Appreciate that not all functions can be given with expressions.
  7. Comprehend that nonlinear behaviour arises when the growth rate is not constant.
  8. Be able to reason graphically

Assessment Methods

The final mark for this unit is obtained from the following contributions:

  • 30% from completed problem sets (best 8 out of 10)
  • 10% from a course project
  • 10% from a written, open book examination in January
  • 50% from a written, open book examination in April
Please note that the use of calculators of the approved type will be allowed in the exams.

Award of Credit Points

You will pass the unit with a mark of 40 or more. You will be awarded 10 credit points by passing the unit. Note that this criterion is different to other first year units, due to the large coursework component.

Transferable Skills

  • Increased understanding of the relationship between mathematics and the “real world” (meaning the physical, biological, economic, etc., systems).
  • Development of problem-solving and analytical skills.
  • Familiarisation with computer tools for mathematics.

Texts

The text for this course is:

“Elementary Mathematical Modeling”, James Sandefur. Thomson, 2003.

Syllabus

Some changes to the syllabus may be applied during the course.

Introduction to modeling – Variables: dependent and independent. Developing simple dynamical systems. Working with functions, explicitly or implicitly. Solving problems with an unknown parameter.

Analysis of dynamical systems – Qualitative behavior of functions satisfying dynamical systems. Solving linear systems of equations to find equilibrium values. The difference between constant and proportional change. Start understanding stability.

Function approach – Develop linear and exponential functions that satisfy dynamical systems. Interpret linear and exponential relationships. Fit data with appropriate functions and interpret results.

Higher order dyamical systems – Counting arguments. Basic analysis of higher order dynamical systems. Application: the study of a national economy.

Nonlinear systems – Introduction to nonlinear behavior. Using graphical tools to interpret rates and analyze nonlinear systems. Finding equilibrium and determining stability.

Population dynamics – Developing reasonable growth rate functions. Reasoning graphically. Interactions, harvesting: considering an optimization problem in the management of resources.

Genetics – Applying all the techniques learned to the study of population genetics, especially evolution resulting from mutation and selection.

Fractals and chaos (optional) – Introduction to self-similarity. Fractals and dimension. Introduction to chaos.