Math for Scientists: Refreshing the Essentials
Book Preface
Almost every student or scientist will at some point run into mathematical formulas or ideas in scientific 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 understanding mathematical symbols and formulas to using complex numbers, dealing with equations involved in calculating medication equivalents, applying the General Linear Model (GLM) used in, e.g., neuroimaging analysis, finding the minimum of a function, applying independent component analysis, or choosing the best filtering approach. In this book we explain the theory behind many of these mathematical ideas and methods and provide readers with the tools to better understand them. We revisit highschool mathematics and extend and relate them to the mathematics you need to understand and apply the math you may encounter in the course of your research. In addition, this book teaches you to understand the math and formulas in the scientific papers you read. To achieve this goal, each chapter mixes theory with practical penandpaper exercises so you (re)gain experience by solving math problems yourself. To provide context, clarify the math, and help readers apply it, each chapter contains realworld and scientific examples. We have also aimed to convey an intuitive understanding of many abstract mathematical concepts.
This book was inspired by a lecture series we developed for junior neuroscientists with very diverse scientific backgrounds, ranging from psychology to linguistics. The initial idea for this lecture series was sparked by a PhD student, who surprised Dr. ĆurčicBlake by not being able to manipulate an equation that involved exponentials, even though she was very bright. Initially, the PhD student even sought help from a statistician who provided a very complex method to calculate the result she was looking for, which she then implemented in the statistical package SPSS. Yet, simple penandpaper exponential and logarithm arithmetic would have solved the problem. Asking around in our departments showed that the problem this particular PhD student encountered was just an example of a more widespread problem and it turned out that many more junior (as well as senior) researchers would be interested in a refresher course about the essentials of mathematics. The first run of lectures in 2014 got very positive feedback from the participants, showing that there is a need for mathematics explained in an accessible way for a broad scientific audience and that the authors’ approach provided that. Since then, we have used our students’ feedback to improve our approach and this book and its affordable paperback format now make this approach to refreshing the ‘math you know you knew’ accessible for a wide readership.
Instead of developing a completely new course, we could have tried to build our course on an existing introductory mathematics book. And of course there are ample potentially suitable mathematics books around. Yet, we find that most are too difficult when you are just looking for a quick introduction to what you learned in high school but forgot about. In addition, most mathematics books that are aimed at bachelorandup students or nonmathematician researchers present mathematics in a mathematical way, with strict rigor, forgetting that readers like to gain an intuitive understanding and ascertain the purpose of what they are learning. Furthermore, many students and researchers who did not study mathematics can have trouble reading and understanding mathematical symbols and equations. 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 the actual mathematics becomes more understandable. With our own firm backgrounds in mathematics (Prof. Maurits) and physics (Dr. ĆurčicBlake) and our working experience and collaborations in the fields 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.
We envisage that undergraduate students and scientists (from PhD students to professors) in disciplines that build on or make use of mathematical principles, such as neuroscience, biology, psychology, or economics, would find this book helpful. The book can be used as a basis for a refresher course of the essentials of (mostly highschool) mathematics, as we use it now. It is also suited for selfstudy, 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. In those cases where earlier discussed topics are needed, we refer to them.
We owe gratitude to several people who have helped us in the process of writing this book. First and foremost, we would like to thank the students of our refresher course for their critical but helpful feedback. Because they did many exercises in the book first, 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 you Dr. Cris Lanting, Dr. Jan Bernard Marsman, and Dr. Remco Renken. Professor Arthur Veldman critically proofread several chapters, which helped incredibly in, especially, clarifying some (too) complicated examples.
Dr. ĆurčicBlake thanks her math school teachers from Tuzla, whom she appreciates and always had a good understanding with. While the highschool math was very easy, she had to put some very hard work in to grasp the math that was taught in her studies of physics. This is why she highly values Professor Milan Vujičic (who taught mathematical physics) and Professor Enes Udovičic (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 giving her ideas for the book and Prof. Maurits for doing the majority of work for this book. Her personal thanks go to her parents Branislav and Spasenka, who always supported her, 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 master students and PhD students, finding 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 supporting her to study mathematics and become the person and researcher she is now, 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 your questions, and that it may help you in your scientific endeavors.
Groningen, The Netherlands Natasha Maurits
Groningen, The Netherlands
April 2017
Branislava ĆurčicBlake
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