Advanced Calculus MATH20472

Course Overview

This Jupyterbook provides the course notes for Advanced Calculus.

Part I - PDEs and Fourier Series

Chapter 1 Introduction

• Initial and boundary value problems (IBVP)
• ODE Revision
• Introduction to partial differential equations (PDEs)
• The equations we shall study

Chapter 2 Fourier Series

• Periodic, even and odd functions
• Fourier series for functions of period $2\pi$
• Convergence of Fourier series
• Functions of any period

Chapter 3 The heat equation

• Derivation in one space dimension
• Heat conduction in a finite rod
• Initial and boundary value problem
• Uniquness (optional)
• Non-zero steady state (optional)

Chapter 4 The wave equation

• Derivation in 1 space dimension
• Normal modes of vibration for a finite string
• Initial and boundary value problems for finite strings
• Uniqueness of an IBVP for a finite string (optional)
• The general solution of the wave equation
• Waves on infinite strings: D'Alembert's formula

Chapter 5 Laplace's equation in the plane

• BVP in cartesian coordinates
• BVP in polar coordinates (optional)
• Uniqueness (optional)

Part II Complex Analysis

Chapter 1 Prerequisites

• Revision of complex numbers
• Complex valued functions

Chapter 2 Differentiation

• Concsequences/uses of the Cauchy-Riemann equations

Chapter 3 Singularities

• Poles
• Removable singularities
• essenital singularities
• singularities at infinity (optional)

Chapter 4 Integration

• Preliminary definitions
• The fundamental theorem of contour integration
• Cauchy's integral theorem
• Cauchy's integral formula
• The annulus (or deformation) theorem

Chapter 5 Taylor and Laurent series

• Taylor series for complex valued functions
• Laurent series

Chapter 6 Residues in complex variable analysis

• Cauchy's residue theorem
• Calculating residues
• Evaluation of real integrals

References

[1] Jordan D.W., Smith P. Mathematical techniques: an introduction for the engineering, physical, and mathematical sciences (2008)

[2] Nagle, R. K., Saff E. B., Snider A. D. Fundamentals of differential equations and boundary value problems (2017)

[3] Kwok Y. K. Applied complex variables for scientists and engineers

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Supplementary Video Lectures

3Blue1Brown: But what is a partial differential equation? YouTube. https://www.youtube.com/watch?v=ly4S0oi3Yz8&t=316s

3Blue1Brown: But what is a Fourier series? From heat flow to drawing circles. YouTube. https://www.youtube.com/watch?v=r6sGWTCMz2k

3Blue1Brown: Complex numbers fundamentals. YouTube. https://www.youtube.com/watch?v=5PcpBw5Hbwo

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Popular Press Reading

Strogatz, S., 2019. Infinite powers: How calculus reveals the secrets of the universe. Houghton Mifflin Harcourt.