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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinany otherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilaror dissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Almosteverystudentorscientistwillatsomepointrunintomathematicalformulasorideas fi 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 fi Model (GLM) used in, e.g., neuroimaging analysis, nding the minimum of a function, fi 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 fi understandthemathandformulasinthescienti cpapersyouread.Toachievethisgoal,each chapter mixes theory with practical pen-and-paper exercises so you (re)gain experience by solvingmathproblemsyourself.Toprovidecontext,clarifythemath,andhelpreadersapply fi it,eachchaptercontainsreal-worldandscienti cexamples.Wehavealsoaimedtoconveyan intuitive understanding of many abstract mathematical concepts. Thisbookwasinspiredbyalectureserieswedevelopedforjuniorneuroscientistswithvery fi 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 fi 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 fi ’ 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 fi 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 fi 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, fi 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 fi 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 fi 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 fi 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 fi 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 ix x Contents 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 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 fi 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 fi 3.3 Trigonometric Functions and Their Complex De nitions. . . . . . . . 68 ’ 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 fi 6.2 Intuitive De nition of Limit. . . . . . . . . . . . . . . . . . . . . . 166 6.3 Determining Limits Graphically. . . . . . . . . . . . . . . . . . . . . 167 6.4 Arithmetic Rules for Limits. . . . . . . . . . . . . . . . . . . . . . . 169
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