How to Derive a Formula Volume 2: Further Analytical Skills and Methods for Physical Scientists

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Will artificial intelligence make scientific formulae redundant by eventually solving all current and future physical problems? The authors of this book would argue that there is still a vital role for humans to play in making sense of the laws of nature. To derive a formula one follows a series of steps, only the last of which is to check that the result is correct. The book is about unravelling this machinery. Mathematics is the 'queen of all sciences', but students encounter many obstacles in learning the subject: familiarization with the proofs of hundreds of theorems, mysterious symbols, and technical routines for which the usefulness is not obvious upfront. Learners could lose motivation, not seeing the wood for the trees. This two-volume book How to Derive a Formula is an attempt to engage learners by presenting mathematical methods in as simple terms as possible, with more of an emphasis on skills as opposed to technical knowledge. Based on intuition and common sense rather than mathematical rigour, it teaches students from scratch using pertinent examples, many taken from across the physical sciences to demonstrate the application of the methods taught. This book draws on humour and historical facts to provide an interesting new perspective on what a mathematics textbook could be. The two volumes are presented as an ascent to Everest. Volume 1 covered the necessary basics, taking readers from Base Camp to Camps 1 and 2. This volume moves readers from Camp 2 up to Camps 3 and 4, tackling more advanced methods for deriving formulae. Inevitably, Volume 2 requires readers to tackle more challenging terrain than was experienced in Volume 1 and so is targeted at more advanced students. Request Inspection Copy Errata(s) Errata (158 KB) Sample Chapter(s) Preface Chapter 1: Multiple Integration in More Depth Contents: Preface About the Authors From Camp 2: In the Multidimensional and Complex World: Multiple Integration in More Depth Surfaces and Vectors Vector Fields and More Vector Calculus Differential Eigenvalue Equations and Special Functions Coupled Linear Second Order Differential Equations and Higher Order Equations Approximation Techniques for Ordinary Differential Equations Homogeneous Partial Differential Equations Complex Contour Integration More Involved Methods for the Summation of Series Concluding Remarks From Camp 3: From Becoming at Ease with Inhomogeneous Partial Differential Equations to a Glimpse of Integral Equations, Functionals, Variation Calculus, Functional Integration, and a Snapshot of the 'Fractional World': Dealing with Inhomogeneous Partial Differential Equations A Brief Look into the World of Integral Equations Principles of Variational Calculus Path Integrals — A First Look In the Thin Air of the Fractional World Concluding Remarks The Final Ascent to Everest Readership: Undergraduate and graduate students (in any disciplines of exact sciences), and postdoctoral researchers in physical sciences, university lecturers for teaching the material of some of the chapters of the book.

How to Derive a Formula Volume 1: Basic Analytical Skills and Methods for Physical Scientists

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Will artificial intelligence solve all problems, making scientific formulae redundant? The authors of this book would argue that there is still a vital role in formulating them to make sense of the laws of nature. To derive a formula one needs to follow a series of steps; last of all, check that the result is correct, primarily through the analysis of limiting cases. The book is about unravelling this machinery. Mathematics is the 'queen of all sciences', but students encounter many obstacles in learning the subject — familiarization with the proofs of hundreds of theorems, mysterious symbols, and technical routines for which the usefulness is not obvious upfront. Those interested in the physical sciences could lose motivation, not seeing the wood for the trees. How to Derive a Formula is an attempt to engage these learners, presenting mathematical methods in simple terms, with more of an emphasis on skills as opposed to technical knowledge. Based on intuition and common sense rather than mathematical rigor, it teaches students from scratch using pertinent examples, many taken across the physical sciences. This book provides an interesting new perspective of what a mathematics textbook could be, including historical facts and humour to complement the material. Errata(s) Errata (756 KB) Sample Chapter(s) Preface Chapter 1: Essential Functions Request Inspection Copy Contents: Preface Introduction From Base Camp — Understanding Functions and Variables: The First Stage: Essential Functions Polynomial Expansions: When They Work and When They Don't Limits, Differentiation and Integration The Way to Check Yourself: Analysis of Limiting Cases Definite Integrals as Functions Probability Distribution Functions, and Filter Functions as Limiting Cases Vectors and Introduction to Vector Calculus Understanding Sequences and Series Complex Numbers Dimensionality and Scaling Concluding Remarks Problems From Camp 1: Deeper Understanding of Functions and Solving Equations: Introduction to Functions of Two or More Variables Fourier Series and Integrals Linear Equations and Determinants Matrices and Symmetry Solving Nonlinear Equations, Algebraic and Transcendental Introduction to Ordinary Differential Equations Further Methods for Evaluating the Integrals and the Gamma Function Functions of a Complex Variable Concluding Remarks Problems Instructions to Access the Outlines of Solutions Readership: Advanced and enthusiastic school students preparing for universities, specializing in science — A-level (UK), Abitur (Germany), Lycée (France), high school (USA) and alike; teachers and tutors; undergraduate students; university lecturers.