AEM-ADV03-Introductory-Mathematics.pdf - AEM ADV03...

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AEM ADV03 Introductory Mathematics 2019/2020
2 Table of Contents: Introductory Mathematics .......................................................................................................................................... 4 1 Function Expansion & Transforms .......................................................................................................................... 7 1.1 Power Series ..................................................................................................................................................... 7 1.1.1 Taylor Series .............................................................................................................................................. 7 1.1.2 Fourier Series ........................................................................................................................................... 12 1.1.3 Complex Fourier series ............................................................................................................................ 14 1.1.4 Termwise Integration and Differentiation ............................................................................................... 15 1.1.5 Fourier series of Odd and Even functions ........................................................................................ 17 1.2 Integral Transform .......................................................................................................................................... 19 1.2.1 Fourier Transform ................................................................................................................................... 20 1.2.2 Laplace Transform ................................................................................................................................... 22 2. Vector Spaces, vector Fields & Operators ............................................................................................................ 25 2.1 Scalar (inner) product of vector fields ............................................................................................................ 26 2.1.1 L p norms .................................................................................................................................................. 27 2.2 Vector product of vector fields ....................................................................................................................... 29 2.3 Vector operators ............................................................................................................................................. 30 2.3.1 Gradient of a scalar field ......................................................................................................................... 30 2.3.2 Divergence of a vector field .................................................................................................................... 33 2.3.3 Curl of a vector field ............................................................................................................................... 35 2.4 Repeated Vector Operations – The Laplacian ................................................................................................ 37 3. Linear Algebra, Matrices & Eigenvectors ............................................................................................................ 42 3.1 Basic definitions and notation ........................................................................................................................ 42 3.2 Multiplication of matrices and multiplication of vectors and matrices .......................................................... 44 3.2.1 Matrix multiplication ............................................................................................................................... 44 3.2.2 Traces and determinants of square Cayley products ............................................................................... 45 3.2.3 The Kronecker product ............................................................................................................................ 45 3.3 Matrix Rank and the Inverse of a full rank matrix ......................................................................................... 47 3.3.1 Full Rank matrices ................................................................................................................................... 47 3.3.2 Solutions of linear equations ................................................................................................................... 48 3.3.3 Preservation of positive definiteness ....................................................................................................... 48 3.3.4 A lower bound on the rank of a matrix product ....................................................................................... 49 3.3.5 Inverse of products and sums of matrices ................................................................................................ 49
3 3.4 Eigensystems .................................................................................................................................................. 50 3.5 Diagonalisation of symmetric matrices .......................................................................................................... 53 4. Generalised Vector Calculus – Integral Theorems .............................................................................................. 56 4.1 The gradient theorem for line integral ............................................................................................................ 56 4.2 Green’s Theorem ............................................................................................................................................ 57 4.3 Stokes’ Theorem ............................................................................................................................................. 62 4.4 Divergent Theorem ......................................................................................................................................... 68 5. Ordinary Differential Equations .......................................................................................................................... 71 5.1 First-Order Linear Differential Equations ...................................................................................................... 71 5.2 Second-Order Linear Differential Equations .................................................................................................. 73 5.3 Initial-Value and Boundary-Value Problems ................................................................................................. 77 5.4 Non-homogeneous linear differential equation .............................................................................................. 80 6. Partial Differential Equations .............................................................................................................................. 83 6.1 Introduction to Differential Equations ............................................................................................................ 83 6.2 Initial Conditions and Boundary Conditions .................................................................................................. 83 6.3 Linear and Nonlinear Equations ..................................................................................................................... 84 6.4 Examples of PDEs ......................................................................................................................................... 86 6.5 Three types of Second-Order PDEs ................................................................................................................ 86 6.6 Solving PDEs using Separation of Variables Method ................................................................................... 87 6.6.1 The Heat Equation ................................................................................................................................... 88 6.6.2 The Wave Equation ................................................................................................................................. 95
4 Introductory Mathematics What is Mathematics? Different schools of thought, particularly in philosophy, have put forth radically different definitions of mathematics. All are controversial and there is no consensus. Leading definitions 1. Aristotle defined mathematics as: The science of quantity . In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry. 2. Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields: The science of indirect measurement , 1851. The ``indirectness'' in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly.

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