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 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 a Some Simple Functions: Trigonometic Degrees vs Radians 360 degrees = 2π radians Degrees: θ 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 Approximating sin Approximating sin 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: or or 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 Complex Numbers 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 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 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 ie 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. .