Maths doesn’t really change, at least the stuff we need for the Applied Environ- mental Hydrogeology Course, at the University of Edinburgh. The following notes were prepared during my early postgraduate days for Masters students en- tering hydrogeology, with a desire to refresh their knowledge, catch up where necessary and fill in the blanks. Particularly this course has proved a constant source of reliable rapid relevant look up and training for me. With the permis- sion of the authors, Professor Rudolf Liedl and Professor Peter Dietrich, we can provide the same course material to yourselves to help you get up to speed where you need to. Enjoy. Professor Chris McDermott, Program Director, Applied Environmental Hydrogeology, University of Edinburgh 17.06.2021 Rudolf Liedl Peter Dietrich Mathematical Methods Lecture Notes 4th ed., October 2004 Liedl / Dietrich Mathematical Methods Table of Contents List of Figures .......................................................................................................................... 4 List of Tables ............................................................................................................................ 5 1 Introduction ...................................................................................................................... 8 2 Statistics ............................................................................................................................. 9 2.1 Descriptive Statistics of One Property ....................................................................... 9 2.1.1 Description Based on a Sorted Data Set ............................................................. 9 2.1.2 Means, Variances and Deviations .................................................................... 10 2.1.3 Frequency Distributions ................................................................................... 11 2.1.4 Comparing Frequency Distributions ................................................................ 13 2.2 Descriptive Statistics of Several Properties .............................................................. 14 2.2.1 Pictorial Representations .................................................................................. 14 2.2.2 Statistical Measures .......................................................................................... 15 2.3 Regression Analysis ................................................................................................. 15 3 Differentiation ................................................................................................................. 17 3.1 Definitions ................................................................................................................ 17 3.2 Rules of Differentiation ............................................................................................ 19 3.3 Higher-order Derivatives .......................................................................................... 20 3.4 Taylor Series ............................................................................................................. 21 4 Integration ....................................................................................................................... 24 4.1 Definitions ................................................................................................................ 24 4.2 Rules of Integration .................................................................................................. 25 4.3 Improper Integrals .................................................................................................... 28 4.4 Applications .............................................................................................................. 31 5 Ordinary Differential Equations ................................................................................... 36 5.1 Definitions and Geometrical Interpretation .............................................................. 36 5.2 Variation of Constants .............................................................................................. 38 5.3 Separation of Variables ............................................................................................ 40 5.4 Deriving and Solving an ODE Model for Contaminant Transport in Soil ............... 42 5.5 Modelling 1D Groundwater Flow in a Heterogeneous Aquifer ............................... 47 6 Vectors and Geometry ................................................................................................... 52 6.1 Scalars and Vectors .................................................................................................. 52 6.1.1 Vector Function of a Scalar Variable ............................................................... 53 6.1.2 Comparison of Vectors ..................................................................................... 53 6.2 Basic Laws of Vector Calculus ................................................................................ 53 6.2.1 Addition and Subtraction .................................................................................. 53 6.2.2 Multiplication of a Vector by a Scalar ............................................................. 54 6.2.3 Linear Combination of Vectors and Decomposition ........................................ 54 6.3 Multiplication of Vectors ......................................................................................... 55 6.3.1 Scalar Product (Dot Product) ........................................................................... 55 6.3.2 Vector Product (Cross Product) ....................................................................... 56 6.3.3 Multiple Products of Vectors ........................................................................... 57 6.4 Applications of Vectors in Geometry ....................................................................... 58 6.4.1 Applications of Vectors for Geometrical Calculations .................................... 58 6.4.2 Equations of a Straight Line ............................................................................. 59 6.4.3 Equations of a Plane ......................................................................................... 62 2 Liedl / Dietrich Mathematical Methods 7 Matrices and Determinants ........................................................................................... 64 7.1 Definition of a Matrix ............................................................................................... 64 7.2 Matrix Algebra ......................................................................................................... 65 7.3 Determinants ............................................................................................................ 66 7.3.1 Calculation for 2x2- and 3x3-Matrices ............................................................ 67 7.3.2 Calculation Using Cofactors ............................................................................ 67 7.3.3 Calculation Rules for Determinants ................................................................. 68 7.3.4 Use of Determinants in Vector Algebra and Geometry ......................................... 68 7.4 Special Types of Matrices ........................................................................................ 69 8 Solving Linear Equation Systems ................................................................................. 71 8.1 Definition of Linear Equation Systems .................................................................... 71 8.2 Existence of Solutions .............................................................................................. 72 8.3 Use of Matrix Calculus ............................................................................................. 72 8.3.1 With the Inverse ............................................................................................... 72 8.3.2 Cramer's Rule ................................................................................................... 73 8.3.3 With Transformation Matrices ......................................................................... 74 8.4 Gaussian Algorithm .................................................................................................. 75 8.5 Homogeneous and Inhomogeneous Systems ........................................................... 77 9 Coordinate Transformations ......................................................................................... 80 9.1 Definition and Types of Coordinate Systems ........................................................... 80 9.2 Polar and Cylindrical Coordinates ........................................................................... 80 9.3 Spherical Coordinates ............................................................................................... 81 9.4 Transformation of Parallel Coordinate Systems ...................................................... 82 9.4.1 2D Transformations .......................................................................................... 82 9.4.2 Homogeneous Coordinates and Matrix Representation of 2DTransformations .......................................................................................................................... 83 9.4.3 Homogeneous Coordinates and Matrix Representation of 3D Transformations .......................................................................................................................... 84 9.4.4