The Entirety of Mathematics in Two Hours

“Do not worry about your problems with mathematics, I assure you mine are far greater”

- Albert Einstein Some Simple Functions: Polynomials

y=x

Some Simple Functions: Polynomials

Simultaneous Equations

Here we have two equations and require they are both true simultaneously.

Function Notation

To show that y depends on x we write:

The name The variable it of the function depends on

If a function depends on more variables we write

Some Simple Functions: Trigonometic

c b

Some Simple Functions: Trigonometic

Degrees vs Radians

360 degrees = 2π radians

Degrees:

Radians:

Everything is just a little bit simpler with radians Some Simple Functions: Trigonometic

Some Simple Functions: Exponential

Defining Functions

● sin, cos and exp difficult to evaluate

● Polynomials easy to evaluate

● Define special functions in terms of a polynomial 'expansion'

Approximating sin

Defining Special Functions

Summation Notation

We call this a series – a list of numbers added together

The fourth term The first term The third term The second term

In this case, the term and there are 4 terms.

Summation Notation

What we are We write summing to

The general term

What we are Index we are summing from summing over

“Sum `n' from 1 to 4”

Summation Notation

The `form' of the general term is clearly

So,

Summation Notation

Complex Numbers

Square roots of positive numbers are well defined:

But what about negative numbers?

Complex Numbers

Let us invent a new number:

Note: this is sometimes called j

Complex Numbers

We can now find the square root of any negative number:

Complex Numbers

We move from a number line to a number plane:

An Argand Diagram. Complex Numbers

- real part

- imaginary part

Complex Numbers

We have the remarkable result that:

This is Euler's Formula

The exponential function is closely linked to the trigonometric functions

Leonard Euler

● Functional analysis ● Mechanics ● Fluid Dynamics ● Astronomy ● Optics ● ...

Matrices and Vectors

We can list (numeric) information as a vector

- a row vector

- a column vector

We denote a vector as v or v e.g.

Matrices and Vectors

We can also store information in a matrix

Or more generically,

Matrices and Vectors

We can multiply vectors and matrices.

We multiply each row of the matrix by the vector in turn

Matrices and Vectors

Another example:

This is the Identity Matrix, denoted I. Matrices and Vectors

In general,

Matrices and Vectors

Generalising even further, we have

which have elements , and .

And we have the formula:

Matrices and Vectors

We can write simultaneous equations in this form:

is equivalent to

or more generically

Matrices and Vectors

If we have matrix A, we define its inverse, A-1, as

We can then rewrite the system of equations:

So we can thus calculate x

Calculus

Isaac Newton Gottfried Wilhelm von Leibniz Calculus

Denounced by Bishop George Berkeley in `The Analyst: A discourse addressed to an infidel mathematician' Described calculus as `the ghosts of departed quantities'

Given a firmer footing by Cauchy and Weierstrass Berkeley

Differentiation

How does a function y vary with x? ie, given a change in x of Δx what is Δy, the corresponding change in y?

Differentiation

For a straight line:

The gradient is Differentiation

A curve can be approximated with a series of lines, each of length

The shorter the lines, the more accurate approximation. It becomes precise as

Straight Line Approximation

We can approximate a circle with a series of straight-sided polygons

4 sides 5 6 8

12 20

Differentiation

Differentiating

Differentiation

Differentiating

So, Differentiation

Integration

What is the area under a curve?

` `

The narrower the rectangles, the more accurate approximation. It becomes precise as Integration

Integration

The Fundamental Theorem of Calculus

For any function: Let us define another funtion, , such that

is the integral of

The Fundamental Theorem of Calculus

Then, we have the relation:

Integration and differentiation are the opposite of each other!

Example of Calculus distance, x

time, t

How fast distance changes (with time) is speed.

Example of Calculus

speed, v

time, t

distance = speed x time = area under the graph

Calculus Notation

A Violin String

Generally one doesn't here a single mode

A superposition of modes occurs instead This adds to the `texture' of the sound

Combinations of modes are simply added

A Violin String

Mode 1

Mode 2

Mode 1 + Mode 2

A Violin String

Fourier Series

By adding together and

Josef Fourier in various ratios we can approximate any* function by it's Fourier Series.

*well, almost any

Fourier Series

For example,

Fourier Series

A Fourier Series is always periodic, so must repeat every 2π

Fourier Series

The general form of a Fourier series is

But how do we find the coefficients and ?

Fourier Series

There is another form of the Fourier series...

where

Fourier Transforms (kind of)

Vibration-o-metre

Plate

Piston

What is driving the plate?

Fourier Transforms (kind of)

Create a Fourier Series for the `time signal', .

We find that all the coefficients are 0, except for:

So the three pistons are vibrating with frequencies

1 4 10 Differential Equations

Population Growth: Number of babies is proportional to number of adults

provides the time scale – a small value gives a small growth rate

We also require an initial condition,

Differential Equations

time

Differential Equations

● Exponential growth reasonable if there is no limitation on resources: eg Bacteria in a petri dish

● If resources are limited, we expect there to be a reduction in growth as we approach a limiting maximum population.

Differential Equations

The logistic equation:

It is nonlinear

Differential Equations

So, for small P we have unrestrained growth but for larger P it is restricted Differential Equations

Levels off near max value

Exponential Growth

time

Differential Equations

Overpopulation

Levels off near max value

time

Differential Equations

● This is an ordinary differential equation – ODE We have only one dependent variable (P) and one independent variable (t)

● What if our function depended on two variables? eg time and location

● We will create a partial differential equation - PDE

Multivariable Calculus

For a function of one variable, we ask how does y(x) vary with respect to x

For a function of two variables, we ask how does y(x,t) vary with respect to either x or t.

So, conceptually it is just the same

Differential Equations The Heat Equation

- the internal heat density

- the heat flux

Change in total heat = difference in flux

Differential Equations The Heat Equation

- the internal heat density

- the heat flux

A conservation law

Differential Equations The Heat Equation ● Fick's Law: The flux of heat is proportional to the temperature gradient

So,

Adolf Fick The Heat Equation Differential Equations

Differential equations must be solved subject to given conditions. eg the population equation required an initial condition

Generally, if you have a time derivative you need a temporal condition. If you have a spatial derivative you need a spatial condition. eg we solve the heat equation subject to the initial condition

and boundary condition

Differential Equations

We put a cake in the oven:

The cake has an initial temperature - the initial condition The oven is a constant temperature - the boundary condition

Differential Equations

The conditions are just as important as the equation itself

e.g. For the logistic equation, and exhibit completely different behaviour.

Likewise a glass of water put in the freezer and a glass of water put in the oven obey the same equation but behave completely differently.