Applied Mathematics at Oxford
Christian Yates
Centre for Mathematical Biology
Mathematical 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:
Experiment
Equipment:
‣ 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
Kepler
‣ Not bad for a 400 year old piece of maths.
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
=
Fourier series
‣ 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.
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/~dacheson/
mechanics.html
‣ Centre for Mathematical Biology
‣ http://www.maths.ox.ac.uk/groups/math
ematical-biology/
‣ My web page
‣ http://people.maths.ox.ac.uk/yatesc/