# Math for Scientists, Refreshing the Essentials by Natasha Maurits and Branislava Curcic Blake

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### Math for Scientists, Refreshing the Essentials Contents

1 Numbers and Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . 1
Natasha Maurits
1.1 What Are Numbers and Mathematical Symbols
and Why Are They Used? . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Classes of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Arithmetic with Fractions . . . . . . . . . . . . . . . . . . . 5
1.2.2 Arithmetic with Exponents and Logarithms . . . . . . . . . . . 8
1.2.3 Numeral Systems . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Mathematical Symbols and Formulas . . . . . . . . . . . . . . . . . . 16
1.3.1 Conventions for Writing Mathematics . . . . . . . . . . . . . 17
1.3.2 Latin and Greek Letters in Mathematics . . . . . . . . . . . . 17
1.3.3 Reading Mathematical Formulas . . . . . . . . . . . . . . . . 17
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 20
Overview of Equations, Rules and Theorems for Easy Reference . . . . . . . . 21
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Equation Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Branislava Ćurčic-Blake
2.1 What Are Equations and How Are They Applied? . . . . . . . . . . . . 27
2.1.1 Equation Solving in Daily Life . . . . . . . . . . . . . . . . . 28
2.2 General Definitions for Equations . . . . . . . . . . . . . . . . . . . . 29
2.2.1 General Form of an Equation . . . . . . . . . . . . . . . . . . 29
2.2.2 Types of Equations . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Combining Like Terms . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Simple Mathematical Operations with Equations . . . . . . . . 31
2.4 Solving Systems of Linear Equations . . . . . . . . . . . . . . . . . . . 32
2.4.1 Solving by Substitution . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Solving by Elimination . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Solving Graphically . . . . . . . . . . . . . . . . . . . . . . 38
2.4.4 Solving Using Cramer’s Rule . . . . . . . . . . . . . . . . . . 39
2.5 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 Solving Graphically . . . . . . . . . . . . . . . . . . . . . . 41
2.5.2 Solving Using the Quadratic Equation Rule . . . . . . . . . . . 42
2.5.3 Solving by Factoring . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Rational Equations (Equations with Fractions) . . . . . . . . . . . . . . 46
2.7 Transcendental Equations . . . . . . . . . . . . . . . . . . . . . . . . 47
2.7.1 Exponential Equations . . . . . . . . . . . . . . . . . . . . . 47
2.7.2 Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . 48
2.8 Inequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.8.1 Introducing Inequations . . . . . . . . . . . . . . . . . . . . 50
2.8.2 Solving Linear Inequations . . . . . . . . . . . . . . . . . . . 50
2.8.3 Solving Quadratic Inequations . . . . . . . . . . . . . . . . . 53
2.9 Scientific Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 56
Overview of Equations for Easy Reference . . . . . . . . . . . . . . . . . . . 57
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Natasha Maurits
3.1 What Is Trigonometry and How Is It Applied? . . . . . . . . . . . . . 61
3.2 Trigonometric Ratios and Angles . . . . . . . . . . . . . . . . . . . . 63
3.2.1 Degrees and Radians . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Trigonometric Functions and Their Complex Definitions . . . . . . . . 68
3.3.1 Euler’s Formula and Trigonometric Formulas . . . . . . . . . . 72
3.4 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 An Alternative Explanation of Fourier Analysis: Epicycles . . . . 78
3.4.2 Examples and Practical Applications of Fourier Analysis . . . . . 79
3.4.3 2D Fourier Analysis and Some of Its Applications . . . . . . . . 83
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 89
Overview of Equations, Rules and Theorems for Easy Reference . . . . . . . . 90
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Natasha Maurits
4.1 What Are Vectors and How Are They Used? . . . . . . . . . . . . . . 99
4.2 Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Vector Addition, Subtraction and Scalar Multiplication . . . . . 101
4.2.2 Vector Multiplication . . . . . . . . . . . . . . . . . . . . . 105
4.3 Other Mathematical Concepts Related to Vectors . . . . . . . . . . . . 113
4.3.1 Orthogonality, Linear Dependence and Correlation . . . . . . . 113
4.3.2 Projection and Orthogonalization . . . . . . . . . . . . . . . . 115
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 121
Overview of Equations, Rules and Theorems for Easy Reference . . . . . . . . 121
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Natasha Maurits
5.1 What Are Matrices and How Are They Used? . . . . . . . . . . . . . . 129
5.2 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2.1 Matrix Addition, Subtraction and Scalar
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2.2 Matrix Multiplication and Matrices
as Transformations . . . . . . . . . . . . . . . . . . . . . . . 133
5.2.3 Alternative Matrix Multiplication . . . . . . . . . . . . . . . . 136
5.2.4 Special Matrices and Other Basic Matrix Operations . . . . . . 137
5.3 More Advanced Matrix Operations and Their Applications . . . . . . . . 139
5.3.1 Inverse and Determinant . . . . . . . . . . . . . . . . . . . . 139
5.3.2 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . 145
5.3.3 Diagonalization, Singular Value Decomposition,
Principal Component Analysis and Independent
Component Analysis . . . . . . . . . . . . . . . . . . . . . . 147
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 154
Overview of Equations, Rules and Theorems for Easy Reference . . . . . . . . 155
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6 Limits and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Branislava Ćurčic-Blake
6.1 Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Intuitive Definition of Limit . . . . . . . . . . . . . . . . . . . . . . 166
6.3 Determining Limits Graphically . . . . . . . . . . . . . . . . . . . . . 167
6.4 Arithmetic Rules for Limits . . . . . . . . . . . . . . . . . . . . . . . 169
6.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6 Application of Limits: Continuity . . . . . . . . . . . . . . . . . . . . 172
6.7 Special Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.8 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.9 Basic Derivatives and Rules for Differentiation . . . . . . . . . . . . . . 177
6.10 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 180
6.11 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.12 Differential and Total Derivatives . . . . . . . . . . . . . . . . . . . . 183
6.13 Practical Use of Derivatives . . . . . . . . . . . . . . . . . . . . . . . 184
6.13.1 Determining Extrema of a Function . . . . . . . . . . . . . . 184
6.13.2 (Linear) Least Squares Fitting . . . . . . . . . . . . . . . . . 187
6.13.3 Modeling the Hemodynamic Response
in Functional MRI . . . . . . . . . . . . . . . . . . . . . . . 189
6.13.4 Dynamic Causal Modeling . . . . . . . . . . . . . . . . . . . 190
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 192
Overview of Equations for Easy Reference . . . . . . . . . . . . . . . . . . . 193
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Branislava Ćurčic-Blake
7.1 Introduction to Integrals . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2 Indefinite Integrals: Integrals as the Opposite of Derivatives . . . . . . . 200
7.2.1 Indefinite Integrals Are Defined Up to a Constant . . . . . . . . 200
7.2.2 Basic Indefinite Integrals . . . . . . . . . . . . . . . . . . . . 201
7.3 Definite Integrals: Integrals as Areas Under a Curve . . . . . . . . . . . 203
7.3.1 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.4.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 209
7.4.2 Integration by Substitution . . . . . . . . . . . . . . . . . . . 212
7.4.3 Integration by the Reverse Chain Rule . . . . . . . . . . . . . 215
7.4.4 Integration of Trigonometric Functions . . . . . . . . . . . . . 217
7.5 Scientific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.5.1 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . 219
7.5.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Symbols Used in This Chapter (in Order of Their Appearance) . . . . . . . . . 225
Overview of Equations for Easy Reference . . . . . . . . . . . . . . . . . . . 225
Answers to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231