# Applied Mathematics at Oxford

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24 Οκτ 2013 (πριν από 4 χρόνια και 7 μήνες)

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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

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

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

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/