Applied Mathematics at Oxford
Christian Yates
Centre for Mathematical Biology
Mathematical Institute
Who
am
I?
‣
Completed my
B.A. (Mathematics) and M.Sc. (Mathematical
M
odelling
and
S
cientific
C
omputing) at
the Mathematical Institute as a member of
Somerville
College.
‣
Currently completing my D.Phil. (
M
athematical Biology) in the Centre for
Mathematical Biology as a member of Worcester and St. Catherine’s colleges.
‣
Next year
–
Junior
R
esearch
F
ellow 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=2x10
30
Kg
‣
G=6.67x10

10
m
3
kg

1
s

2
‣
R=1.5x10
11
m
‣
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
F
ourier
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
H
ow
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/
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