Math for scientists : refreshing the essentials

240 Pages·2017·7.31 MB·English

by  Ćurčić-Blake

Math
Natasha Maurits
Branislava Ćurčić-Blake
for Scientists
R E F R E S H I N G T H E E S S E N T I A L S
Math for Scientists
(cid:129) Ć č (cid:1)
Natasha Maurits Branislava ur ic-Blake
Math for Scientists
Refreshing the Essentials
NatashaMaurits BranislavaĆurči(cid:1)c-Blake
DepartmentofNeurology NeuroimagingCenter
UniversityMedicalCenterGroningen UniversityMedicalCenterGroningen
Groningen,TheNetherlands Groningen,TheNetherlands
ISBN978-3-319-57353-3 ISBN978-3-319-57354-0 (eBook)
DOI10.1007/978-3-319-57354-0
LibraryofCongressControlNumber:2017943515
©SpringerInternationalPublishingAG2017
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Preface
Almosteverystudentorscientistwillatsomepointrunintomathematicalformulasorideas
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in scienti c papers that may be hard to understand or apply, given that formal math
education may be some years ago. These math issues can range from reading and under-
standing mathematical symbols and formulas to using complex numbers, dealing with
equations involved in calculating medication equivalents, applying the General Linear
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Model (GLM) used in, e.g., neuroimaging analysis, nding the minimum of a function,
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applying independent component analysis, or choosing the best ltering approach. In this
book we explain the theory behind many of these mathematical ideas and methods and
providereaderswiththetoolstobetterunderstandthem.Werevisithigh-schoolmathemat-
ics and extend and relate them to the mathematics you need to understand and apply the
mathyoumayencounterinthecourseofyourresearch.Inaddition,thisbookteachesyouto
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understandthemathandformulasinthescienti cpapersyouread.Toachievethisgoal,each
chapter mixes theory with practical pen-and-paper exercises so you (re)gain experience by
solvingmathproblemsyourself.Toprovidecontext,clarifythemath,andhelpreadersapply
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it,eachchaptercontainsreal-worldandscienti cexamples.Wehavealsoaimedtoconveyan
intuitive understanding of many abstract mathematical concepts.
Thisbookwasinspiredbyalectureserieswedevelopedforjuniorneuroscientistswithvery
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diversescienti cbackgrounds,rangingfrompsychologytolinguistics.Theinitialideaforthis
Ć č(cid:1)
lecture series was sparked by a PhD student, who surprised Dr. ur ic-Blake by not being
abletomanipulateanequationthatinvolvedexponentials,eventhoughshewasverybright.
Initially,thePhDstudenteven soughthelpfromastatisticianwhoprovidedaverycomplex
method to calculate the result she was looking for, which she then implemented in the
statistical package SPSS. Yet, simple pen-and-paper exponential and logarithm arithmetic
wouldhavesolvedtheproblem.Askingaroundinourdepartmentsshowedthattheproblem
thisparticularPhDstudentencounteredwasjustanexampleofamorewidespreadproblem
anditturnedoutthatmanymorejunior(aswellassenior)researcherswouldbeinterestedin
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a refresher course about the essentials of mathematics. The rst run of lectures in 2014 got
very positive feedback from the participants, showing that there is a need for mathematics
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explained inanaccessible wayforabroad scienti caudience andthat theauthors approach
v
vi Preface
’
providedthat.Sincethen,wehaveusedourstudents feedbacktoimproveourapproachand
this book and its affordable paperback format now make this approach to refreshing the
‘ ’
math you know you knew accessible for a wide readership.
Insteadofdevelopingacompletelynewcourse,wecouldhavetriedtobuildourcourseon
an existing introductory mathematics book. And of course there are ample potentially
fi fi
suitable mathematics books around. Yet, we nd that most are too dif cult when you are
justlookingforaquickintroductiontowhatyoulearnedinhighschoolbutforgotabout.In
addition, most mathematics books that are aimed at bachelor-and-up students or
non-mathematician researchers present mathematics in a mathematical way, with strict
rigor,forgettingthatreadersliketogainanintuitiveunderstandingandascertainthepurpose
of what they are learning. Furthermore, many students and researchers who did not study
mathematics can have trouble reading and understanding mathematical symbols and equa-
tions. Even though our book is not void of mathematical symbols and equations, the
introduction to each mathematical topic is more gradual, taking the reader along, so that
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the actual mathematics becomes more understandable. With our own rm backgrounds in
Ć č(cid:1)
mathematics(Prof.Maurits)andphysics(Dr. ur ic-Blake)andourworkingexperienceand
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collaborations in the elds of biophysical chemistry, neurology, psychology, computer
science, linguistics, biophysics, and neuroscience, we feel that we have the rather unique
combination of skills to write this book.
Weenvisagethatundergraduatestudentsandscientists(fromPhDstudentstoprofessors)
in disciplines that build on or make use of mathematical principles, such as neuroscience,
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biology,psychology,oreconomics,would ndthisbookhelpful.Thebookcanbeusedasa
basisforarefreshercourseoftheessentialsof(mostlyhigh-school)mathematics,asweuseit
now. It is also suited for self-study, since we provide ample examples, references, exercises,
and solutions. The book can also be used as a reference book, because most chapters can be
read and studied independently. Inthose cases where earlier discussed topicsare needed,we
refer to them.
Weowegratitudetoseveralpeoplewhohavehelpedusintheprocessofwritingthisbook.
First and foremost, we would like to thank the students of our refresher course for their
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critical but helpful feedback. Because they did many exercises in the book rst, they also
helped us to correct errors in answers. The course we developed was also partially taught by
other scientists who helped us shape the book and kindly provided some materials. Thank
youDr.CrisLanting,Dr.JanBernardMarsman,andDr.RemcoRenken.ProfessorArthur
Veldman critically proofread several chapters, which helped incredibly in, especially, clarify-
ing some (too) complicated examples.
Ć č(cid:1)
Dr. ur ic-BlakethankshermathschoolteachersfromTuzla,whomsheappreciatesand
alwayshadagoodunderstandingwith.Whilethehigh-schoolmathwasveryeasy,shehadto
putsomeveryhardworkintograspthemaththatwastaughtinherstudiesofphysics.Thisis
č(cid:1)
why she highly values Professor Milan Vuji ic (who taught mathematical physics) and
č(cid:1)
Professor Enes Udovi ic (who taught mathematics 1 and 2) from Belgrade University who
encouraged her to do her best and to learn math.She would like to thank her colleagues for
givingherideasforthebookandProf.Mauritsfordoingthemajorityofworkforthisbook.
Her personal thanks go to her parents Branislav and Spasenka, who always supported her,
Preface vii
her sons Danilo and Matea for being happy children, and her husband Graeme Blake for
enabling her, while writing chapters of this book.
One of the professional tasks Professor Maurits enjoys most is teaching and supervising
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master students and PhD students, nding it very inspiring to see sparks of understanding
and inspiration ignite in these junior scientists. With this book she hopes to ignite a similar
spark of understanding and hopefully enjoyment toward mathematics in a wide audience of
scientists, similar to how the many math teachers she has had since high school did in her.
She thanks her students for asking math questions that had her dive into the basics of
mathematics again and appreciate it once more for its logic and beauty, her parents for
supportinghertostudymathematicsandbecomethepersonandresearchersheisnow,and,
‘ ’
last but not least, Johan for bearing with her through the writing of yet another book and
providing many cups of tea.
Finally, we thank you, the reader, for opening this book in an effort to gain more
understanding of mathematics. We hope you enjoy reading it, that it gives you answers to
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your questions, and that it may help you in your scienti c endeavors.
Groningen, The Netherlands Natasha Maurits
Ć č(cid:1)
Groningen, The Netherlands Branislava ur ic-Blake
April 2017
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 Ćurči(cid:1)c-Blake
2.1 What Are Equations and How Are They Applied?. . . . . . . . . . . . 27
2.1.1 Equation Solving in Daily Life. . . . . . . . . . . . . . . . . 28
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2.2 General De nitions 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
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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
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2.4.4 Solving Using Cramers 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
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2.9 Scienti c 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
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3.3 Trigonometric Functions and Their Complex De nitions. . . . . . . . 68
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3.3.1 Eulers 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
Contents xi
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 Ćurči(cid:1)c-Blake
6.1 Introduction to Limits. . . . . . . . . . . . . . . . . . . . . . . . . 163
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6.2 Intuitive De nition of Limit. . . . . . . . . . . . . . . . . . . . . . 166
6.3 Determining Limits Graphically. . . . . . . . . . . . . . . . . . . . . 167
6.4 Arithmetic Rules for Limits. . . . . . . . . . . . . . . . . . . . . . . 169