This book reviews math topics relevant to non-mathematics students and scientists, but which they may not have seen or studied for a while. These math issues can range from reading mathematical symbols, to using complex numbers, dealing with equations involved in calculating medication equivalents, the General Linear Model (GLM) used in e.g. neuroimaging analysis, finding the minimum of a function, independent component analysis, or filtering approaches. Almost every student or scientist, will at some point run into mathematical formulas or ideas in scientific papers that may be hard to understand, given that formal math education may be some years ago. In this book we will explain the theory behind many of these mathematical ideas and expressions and provide readers with the tools to better understand them. We will revisit high school mathematics and extend and relate this to the mathematics you need to understand the math you may encounter in the course of your research. This book will help you understand the math and formulas in the scientific papers you read. To achieve this goal, each chapter mixes theory with practical pen-and-paper exercises such that you (re)gain experience with solving math problems yourself. Mnemonics will be taught whenever possible. To clarify the math and help readers apply it, each chapter provides real-world and scientific examples. In this new edition, two new chapters covering statistics and differential equations have been added, which have been workshopped in the 'authors' popular lecture series in order to maximize the benefit for readers.
Author(s): Natasha Maurits, Branislava Ćurčić-Blake
Edition: 2
Publisher: Springer
Year: 2023
Language: English
Commentary: TruePDF
Pages: 319
Tags: Popular Science In Mathematics; Popular Science: General; Life Sciences: General; Study And Learning Skills; Science, Multidisciplinary, Mathematics: General
Preface to the Second Edition
Preface to the First Edition
Contents
Abbreviations
1: Numbers and Mathematical Symbols
1.1 What Are Numbers and Mathematical Symbols and Why Are They Used?
1.2 Classes of Numbers
Exercise
1.2.1 Arithmetic with Fractions
Exercise
Exercise
Exercise
1.2.2 Arithmetic with Exponents and Logarithms
Exercise
Exercise
Exercise
1.2.3 Numeral Systems
Exercise
Exercise
1.2.4 Complex Numbers
Exercise
1.2.4.1 Arithmetic with Complex Numbers
Exercise
1.2.4.2 The Polar Form of Complex Numbers
1.3 Mathematical Symbols and Formulas
1.3.1 Conventions for Writing Mathematics
1.3.2 Latin and Greek Letters in Mathematics
1.3.3 Reading Mathematical Formulas
Glossary
1.3 Symbols Used in This Chapter (in Order of Their Appearance)
1.3 Overview of Equations, Rules and Theorems for Easy Reference
1.3 Answers to Exercises
References
Online Sources of Information: Methods
Online Sources of Information: Other
Book
Paper
2: Equation Solving
2.1 What Are Equations and How Are They Applied?
2.1.1 Equation Solving in Daily Life
Example 2.1
Example 2.2
2.2 General Definitions for Equations
2.2.1 General Form of an Equation
2.2.2 Types of Equations
2.3 Solving Linear Equations
2.3.1 Combining Like Terms
Example 2.3
Exercise
2.3.2 Simple Mathematical Operations with Equations
Box 2.1: Useful Arithmetic Rules for Solving Linear Equations
Exercise
2.4 Solving Systems of Linear Equations
Example 2.4
Example 2.5
2.4.1 Solving by Substitution
Example 2.6
Example 2.7
Exercises
2.4.2 Solving by Elimination
Example 2.8
Example 2.9
Example 2.10
Exercise
2.4.3 Solving Graphically
Example 2.11
2.4.4 Solving Using Cramer´s Rule
Box 2.2: Cramer´s Rule for a System of 2 Linear Equations
2.5 Solving Quadratic Equations
Example 2.12
Example 2.13
2.5.1 Solving Graphically
2.5.2 Solving Using the Quadratic Equation Rule
Box 2.3: Quadratic Equation Rule
Example 2.14
Exercise
2.5.3 Solving by Factoring
Example 2.15
Box 2.4: Factor Multiplication Rule
Example 2.16
Example 2.17
Example 2.18
Example 2.16 (continued)
Exercise
2.6 Rational Equations (Equations with Fractions)
Example 2.19
2.7 Transcendental Equations
2.7.1 Exponential Equations
Example 2.20
Example 2.21
Exercise
2.7.2 Logarithmic Equations
Example 2.22
Example 2.23
Exercise
2.8 Inequations
2.8.1 Introducing Inequations
2.8.2 Solving Linear Inequations
Exercise
2.8.3 Solving Quadratic Inequations
Example 2.24
Example 2.25
Exercise
2.9 Scientific Example
Example 2.26
Glossary
2.9 Appendices
Appendix A: Symbols Used in This Chapter (in Order of Their Appearance)
Appendix B: Overview of Equations for Easy Reference
Appendix C: Answers to Exercises
References
Online Sources of Information
Book
Paper
3: Trigonometry
3.1 What Is Trigonometry and How Is It Applied?
3.2 Trigonometric Ratios and Angles
Exercise
3.2.1 Degrees and Radians
Exercises
3.3 Trigonometric Functions and Their Complex Definitions
Exercises
Exercise
Exercises
3.3.1 Euler´s Formula and Trigonometric Formulas
3.4 Fourier Analysis
Box 3.1 Summary of the mathematics of Fourier series and Fourier transform (based on `From Neurology to Methodology and back. ...
3.4.1 An Alternative Explanation of Fourier Analysis: Epicycles
3.4.2 Examples and Practical Applications of Fourier Analysis
Exercise
3.4.3 2D Fourier Analysis and Some of Its Applications
Glossary
3.4 Appendices
Appendix A: Symbols Used in This Chapter (in Order of Their Appearance)
Appendix B: Overview of Equations, Rules and Theorems for Easy Reference
Appendix C: Answers to Exercises
References
Online Sources of Information: Methods
Book
Papers
4: Vectors
4.1 What Are Vectors and How Are They Used?
4.2 Vector Operations
Box 4.1 Properties of Binary Mathematical Operations (Examples)
4.2.1 Vector Addition, Subtraction and Scalar Multiplication
Example 4.1
Exercises
4.2.2 Vector Multiplication
4.2.2.1 Inner Product
Exercises
4.2.2.2 Cross Product
Example 4.2
Exercises
4.3 Other Mathematical Concepts Related to Vectors
4.3.1 Orthogonality, Linear Dependence and Correlation
Exercises
4.3.2 Projection and Orthogonalization
Example 4.3
Example 4.4
Glossary
4.3 Symbols Used in This Chapter (in Order of Their Appearance)
4.3 Overview of Equations, Rules and Theorems for Easy Reference
4.3 Answers to Exercises
References
Online Sources of Information: History
Online Sources of Information: Methods
Papers
5: Matrices
5.1 What Are Matrices and How Are They Used?
5.2 Matrix Operations
5.2.1 Matrix Addition, Subtraction and Scalar Multiplication
Exercises
5.2.2 Matrix Multiplication and Matrices as Transformations
Exercises
5.2.3 Alternative Matrix Multiplication
Exercises
5.2.4 Special Matrices and Other Basic Matrix Operations
Exercises
5.3 More Advanced Matrix Operations and Their Applications
5.3.1 Inverse and Determinant
Box 5.1 Example of Calculating the Inverse of a Matrix
Exercises
Box 5.2 How Discretizing a Partial Differential Equation Can Yield a Sparse Matrix
5.3.2 Eigenvectors and Eigenvalues
Box 5.3 Example of Calculating the Eigenvalues and Eigenvectors of a Matrix
Exercises
5.3.3 Diagonalization, Singular Value Decomposition, Principal Component Analysis and Independent Component Analysis
Box 5.4 Example of SVD of a Real Square Matrix and Its Intuitive Understanding
Exercises
Glossary
5.3 Appendices
Symbols Used in This Chapter (in Order of Their Appearance)
Overview of Equations, Rules and Theorems for Easy Reference
Answers to Exercises
References
Online Sources of Information: History
Online Sources of Information: Methods
Books
Papers
6: Limits and Derivatives
6.1 Introduction to Limits
Example 6.1
Example 6.2
6.2 Intuitive Definition of Limit
Example 6.3
Exercise
6.3 Determining Limits Graphically
Example 6.4
Example 6.5
Example 6.6
6.4 Arithmetic Rules for Limits
Box 6.1: Arithmetic Rules for Limits
Example 6.7
Exercise
6.5 Limits at Infinity
Example 6.8
Example 6.9
Example 6.10
Exercise
6.6 Application of Limits: Continuity
6.7 Special Limits
Box 6.2: Special Limits
6.8 Derivatives
Box 6.3: Alternative Definitions of a Derivative
6.9 Basic Derivatives and Rules for Differentiation
Example 6.11
Example 6.12
Example 6.13
Example 6.14
Example 6.15
Exercise
Example 6.16
Example 6.17
Example 6.18
Exercise
6.10 Higher Order Derivatives
Box 6.4: Higher Order Derivatives
Example 6.19
Exercise
6.11 Partial Derivatives
Example 6.20
Box 6.5: Partial Derivative Notation
Box 6.6: Second Order Partial Derivatives
Example 6.21
Exercise
6.12 Differential and Total Derivatives
Example 6.22
6.13 Practical Use of Derivatives
6.13.1 Determining Extrema of a Function
Box 6.7: Distinguishing Maxima and Minima of a Function
Example 6.23
Example 6.24
Example 6.25
6.13.2 (Linear) Least Squares Fitting
6.13.3 Modeling the Hemodynamic Response in Functional MRI
6.13.4 Dynamic Causal Modeling
Example 6.26
Example 6.27
Glossary
6.13 Symbols Used in This Chapter (in Order of Their Appearance)
6.13 Overview of Equations for Easy Reference
6.13 Answers to Exercises
References
Online Sources of Information
Papers
7: Integrals
7.1 Introduction to Integrals
7.2 Indefinite Integrals: Integrals as the Opposite of Derivatives
Example 7.1
7.2.1 Indefinite Integrals Are Defined Up to a Constant
Example 7.2
Example 7.3
7.2.2 Basic Indefinite Integrals
Example 7.4
Box 7.1 Basic Rules of Integration
Example 7.5
Exercise
7.3 Definite Integrals: Integrals as Areas Under a Curve
Example 7.6
Box 7.2 Important Rules for Definite Integrals
Example 7.7
Example 7.8
Example 7.9
Exercise
7.3.1 Multiple Integrals
Example 7.10
7.4 Integration Techniques
7.4.1 Integration by Parts
Example 7.11
Example 7.12
Example 7.13
Exercise
7.4.2 Integration by Substitution
Example 7.14
Example 7.15
Example 7.16
Example 7.17
Example 7.18
Exercise
7.4.3 Integration by the Reverse Chain Rule
Example 7.19
Exercise
7.4.4 Integration of Trigonometric Functions
Example 7.20
7.5 Scientific Examples
7.5.1 Expected Value
Example 7.21
Example 7.22
7.5.2 Convolution
Example 7.23
Example 7.24
Example 7.25
Example 7.26
Example 7.27
Example 7.28
Example 7.29
Glossary
7.5 Symbols Used in This Chapter (in Order of Their Appearance)
7.5 Overview of Equations for Easy Reference
7.5 Answers to Exercises
References
Online Sources of Information
Books
8: Statistics
8.1 What Is Statistics and How Is It Applied?
Exercise
8.2 Descriptive Statistics
8.2.1 Types of Data
Exercise
8.2.2 Sample and Sampling
8.2.3 Data Visualization
8.2.3.1 Bar Charts and Histograms
8.2.3.2 Scatter Plots and Line Charts
8.2.3.3 Box and Violin Plots
Box 8.1 Quantiles
8.2.4 Summary Statistics
Exercise
Exercise
8.2.5 Data Distribution
Exercise
8.2.5.1 Normal Distribution
Exercise
Exercise
Exercise
Exercise
8.3 Inferential Statistics
8.3.1 Hypothesis Testing
Exercise
8.3.2 Statistical Tests
8.3.3 Drawing Conclusions
Exercise
8.4 Statistical Relationships
Glossary
8.4 Appendices
Appendix A: Overview of Equations, Rules and Theorems for Easy Reference
Appendix B: Answers to Exercises
References
Online Sources of Information
Methods
Misleading Statistics
Data Visualization
Books
Papers
9: Differential Equations
9.1 Introduction to Differential Equations?
9.1.1 What Are Differential Equations?
9.1.2 Differential Equations in Daily Life
Example 9.1
9.1.3 Order and Degree of Differential Equations
9.1.3.1 Order
Exercise
9.1.3.2 Degree
Example 9.2
Example 9.3
Exercise
9.1.4 Ordinary Versus Partial and Linear Versus Non-linear Differential Equations
9.1.4.1 Ordinary Versus Partial Differential Equations
Example 9.4
9.1.4.2 Linear Versus Non-linear Differential Equations
Example 9.5
Exercise
9.2 Solutions of Differential Equations
Exercise
Example 9.6
Example 9.7
Example 9.7. extension
Example 9.8
9.2.1 General Solution Versus Specific Solution
Example 9.9
Exercise
9.2.2 Visualization
9.3 Separable Differential Equations and Their Solutions
Example 9.10
Example 9.11
Exercise
9.4 First and Second Order Linear Differential Equations
9.4.1 Solutions to First Order Homogeneous Differential Equations
Example 9.12
Exercise
9.4.2 Solutions to First Order Non-homogeneous Differential Equations: The Integrating Factor Technique
Example 9.13
Exercise
9.4.3 Solutions to Homogeneous Second Order Linear Differential Equations
Example 9.14
Exercise
9.4.4 Solutions to Non-homogeneous Second Order Linear Differential Equations
Example 9.15
Exercise
9.5 Examples from the Real World
9.5.1 Population Growth
9.5.2 Electrical Circuits
Glossary
9.5 Appendices
Appendix A: Symbols Used in This Chapter (in Order of Their Appearance)
Appendix B: Overview of Equations for Easy Reference
Appendix C: Answers to Exercises
References
Online Sources of Information
Books
Index
