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MM311: Mathematics 4B by Mind Map: MM311: Mathematics 4B
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MM311: Mathematics 4B

0. Probability and Statistics

53.202 Materials

Lecture Notes

Worked Examples

Tutorials and Solutions

Past Papers

Presentation and Summarisation of Data

Hisograms

Stem and Leaf Plots

Box Plots

Good and Bad Presentation

Introduction to Exploratory Data

Lecture Slides, Lecture 1, Lecture 2

RandomVariables

General Results

Discrete and Continuous Random Variables

Expectation

Variance

Moments

Quartiles

Mean and Variance of Linear Combinaions

Lecture Slides, Lecture 5, Lecture 6

Basic Laws

P(A or B)

P(not A)

P(A and B)

P(A|B)

Independence

Bayes Theorem

Lecture Slides, Lecture 4, Lecture 5

Probability Theory

Introduction

Origin of Probability

Assignment of Probability

Disributions, Poisson, Binomial, Geometric, Negative ~Exponential, Nomal

Lecture Slides, Lecture 3, Lecture 7, Lecture 8, Lecture 9

Measures of Spread

Variance

Standard Deviation

Range

Quartiles

Semi-interquartile Range

Lecture Slides, Lecture 6

Measures of Locaion

Mean

Median

Mode

Z-test, samples and confidence

Lecture Slides, Lecture 10, Lecture 11, Lecture 12

Coursework (Semester 2)

Sheet 1

Solutions

Sheet 2

Solutions

Sheet 3

Solutions

1. Fourier Series

Periodic Functions

Extended Functions

Sketching Functions

Odd and Even Functions

Half Range Fourier Series

Fourier Sine Series

Fourier Cosine Series

Continuous Fourier Series

Term-by-Term Differentiation

Complex Exponential Form

Tutorials

Solutions

Worked Examples

2. Partial Differential Equations

Solving Simple PDE's

Change of Variables

Real Examples of PDE's

Heat Equation

Wave Equation

Linearity, Homogeneity and Superposition

Constant Coefficient ODE's

Separation of Variables

Tutorials

Solutions

Worked Examples

3. Complex Analysis

Complex Numbers

Points in the Complex Plane

Complex Functions

Continuity

Analytic Functions

The Cauchy-Riemann Equations

Harmonic Functions

Functions of Complex Variables

Complex Exponential

Complex Trig & Hyperbolic Functions

Complex Logarithm

Line Integrals in the Complex Plane

Parameterising Curves

Cauchy's Integral Theorem

Cauchy's Integral Formula

CIF for Derivatives of Analytic Functions

Tutorials

Solutions (Part 1)

Solutions (Part 2)

4. Techniques of Complex Integration

Taylor and Laurent Series

Poles

Residue Integration

Method

Theorem

Residue at Simple Pole, 1st Method, 2nd Method

Residue at nth Order Pole

Real Integration using Residue Theorem

Integrals of Rational Functions

Improper Integrals

Fourier Integrals

Tutorials

Solutions

5. Fourier Transforms

The Fourier Transform

The Fourier Sine and Cosine Transforms

Properties of the Fourier Transform

Linearity

Derivative

Time-Shift

Frequency-Shift

Duality

Convolution of Two Functions, In Time, In Frequency

Multiplication

Parseval's Identity

Dirac Delta Function

Generalised Fourier Transforms

Application of Residue Integration to Fourier Transforms

Tutorials

Solutions

Sample Papers

Sample Paper 1

Solutions

Sample Paper 2