Math for Scientists - Refreshing the Essentials

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. Th is 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.

In this new edition, two new chapters covering statistics and differential equations have been added, which have been workshopped in the 'authors' popular lecture series in order to maximize the benefit for readers.

Inhaltsverzeichnis:

Preface.- Numbers and mathematical symbols: natural, rational, irrational and complex numbers/complex plane: formula reading, often used symbols in mathematical formulas.- Equations: equalities and inequalities: expansions, series: fractional equations: equation solving techniques: various rules (such as Cramer's rule) to solve equations: introduction to basic functions (e.g. square, square root).- Trigonometry: trigonometric ratios, angles: trigonometric functions (sin, cos, tan) and their complex definitions: epicycles: Fourier series and transform.- Vectors: geometric interpretation of vectors: vector addition/subtraction, scalar multiplication: projections: inner product (including related aspects such as correlation, independence and orthogonality).- Matrices: basic matrix manipulations e.g. multiplication and inversion with examples such as the Jacobian, affine transformation, and rotation: Principal Component analysis in matrix notation.- Differentiation: limits and inf inity: continuity of a function: the differential: basic differentiation rules: partial differential equations: introduction to dynamic systems.- Integration: explanation in terms of antiderivatives and area under the curve: basic integration rules: convolution.- statistics.- differential equations.