Applied Mathematics at Oxford

Applied Mathematics at OxfordChristian YatesCentre for Mathematical BiologyMathematical Institute

Who am I?‣ Completed my B.A. (Mathematics) and M.Sc. (Mathematical

Modelling and Scientific Computing) at the Mathematical Institute as a member of Somerville College.

‣ Currently completing my D.Phil. (Mathematical Biology) in the Centre for Mathematical Biology as a member of Worcester and St. Catherine’s colleges.

‣ Next year – Junior Research Fellow at Christ Church college.‣ Research in cell migration, bacterial motion and locust motion.‣ Supervising Masters students.

‣ Lecturer at Somerville College‣ Teaching 1st and 2nd year tutorials in college.

Outline of this talk‣ The principles of applied mathematics

‣ A practical example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves on strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves on strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Calculus of variations‣ Mathematical Biology

‣ Reasons to study mathematics

Principles of applied mathematics‣ Start from a physical or “real world” system‣ Use physical principles to describe it using mathematics

‣ For example, Newton’s Laws‣ Derive the appropriate mathematical terminology

‣ For example, calculus‣ Use empirical laws to turn it into a solvable mathematical problem

‣ For example, Law of Mass Action, Hooke’s Law‣ Solve the mathematical model

‣ Develop mathematical techniques to do this‣ For example, solutions of differential equations

‣ Use the mathematical results to make predictions about the real world system

Simple harmonic motion‣ Newton’s second law

‣ Force = mass x acceleration

‣ Hooke’s Law‣ Tension = spring const. x extension

‣ Resulting differential equation

simple harmonic motion‣ Re-write in terms of the displacement from equilibrium

which is the description of simple harmonic motion‣ The solution is

with constants determined by the initial displacement and velocity‣ The period of oscillations is

Putting maths to the test: Prediction‣ At equilibrium (using Hooke’s law T=ke):

‣ Therefore:

‣ So the period should be:

ExperimentEquipment:

‣ Stopwatch‣ Mass‣ Spring‣ Clampstand‣ 1 willing volunteer

‣ Not bad but not perfect‣ Why not?

‣ Air resistance‣ Errors in measurement etc‣ Old Spring‣ Hooke’s law isn’t perfect etc

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves on strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

Celestial mechanics‣ Newton’s 2nd Law

‣ Newton’s Law of Gravitation

‣ The position vector satisfies the differential equation

Solution of this equation confirms Kepler’s Laws

How long is a year?‣ M=2x1030 Kg

‣ G=6.67x10-10 m3kg-1s-2

‣ R=1.5x1011m

‣ Not bad for a 400 year old piece of maths.

Kepler

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves on strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

Waves on a string‣ Apply Newton’s Law’s to each

small interval of string...

‣ The vertical displacement satisfies the partial differential equation

‣ Known as the wave equation‣ Wave speed:

Understanding music‣ Why don’t all waves sound like this?

‣ Because we can superpose waves on each other

=

‣ By adding waves of different amplitudes and frequencies we can come up with any shape we want:

‣ The maths behind how to find the correct signs and amplitudes is called Fourier series analysis.

Fourier series

More complicated wave forms‣ Saw-tooth wave:

‣ Square wave:

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves of strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

Fluid mechanics‣ Theory of flight - what causes the lift on an aerofoil?

‣ What happens as you cross the sound barrier?

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves of strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

Classical mechanics‣ Can we predict the motion of a double

pendulum?

‣ In principle yes.

‣ In practice, chaos takes over.

Outline of this talk‣ The principles of applied

mathematics‣ A simple example

‣ Mods applied mathematics (first year)‣ Celestial mechanics‣ Waves of strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Mathematical Biology

‣ Reasons to study mathematics

How we do mathematical biology?‣ Find out as much as we can

about the biology‣ Think about which bits of

our knowledge are important

‣ Try to describe things mathematically

‣ Use our mathematical knowledge to predict what we think will happen in the biological system

‣ Put our understanding to good use

Mathematical biology

Locusts

Switching behaviour‣ Locusts switch direction periodically

‣ The length of time between switches depends on the density of the group

30 Locusts 60 Locusts

Explanation - Cannibalism

Outline of this talk‣ The principles of applied mathematics

‣ A simple example‣ Mods applied mathematics (first year)

‣ Celestial mechanics‣ Waves on strings

‣ Applied mathematics options (second and third year)‣ Fluid mechanics‣ Classical mechanics‣ Calculus of variations‣ Mathematical Biology

‣ Reasons to study mathematics

Why mathematics?‣ Flexibility - opens many doors ‣ Importance - underpins science

‣ Ability to address fundamental questions about the universe‣ Relevance to the “real world” combined with the beauty of

abstract theory‣ Excitement - finding out how things work‣ Huge variety of possible careers‣ Opportunity to pass on knowledge to others

Me on Bang goes the theory

I’m off to watch Man City in the FA cup final

Further information‣ Studying mathematics and joint

schools at Oxford‣ http://www.maths.ox.ac.uk

‣ David Acheson’s page on dynamics‣ http://home.jesus.ox.ac.uk/~dache

son/mechanics.html‣ Centre for Mathematical Biology

‣ http://www.maths.ox.ac.uk/groups/mathematical-biology/

‣ My web page‣ http://people.maths.ox.ac.uk/yates

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