Math for Scientists: Refreshing the Essentials

by Branislava Ćurčić-Blake, Natasha Maurits

Length: 246 pages
Edition: 1
Language: English
Publisher: Springer
Publication Date: 2017-09-07
ISBN-10: 3319573535
ISBN-13: 9783319573533
Sales Rank: #2117435 (See Top 100 Books)

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.

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