Math for Scientists: Refreshing the Essentials

https://lavozdelascostureras.com/ajs9tyd Preface 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 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: Methods Online Sources of Information: Others Books Papers 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 Symbols Used in This Chapter (in Order of Their Appearance) Overview of Equations for Easy Reference Answers to Exercises References Online Sources of Information Books Papers 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 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: Methods Books 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 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 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 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 Symbols Used in This Chapter (in Order of Their Appearance) Overview of Equations for Easy Reference 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 Symbols Used in This Chapter (in Order of Their Appearance) Overview of Equations for Easy Reference Answers to Exercises References Online Sources of Information Books Index