Understanding Equations
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Thinking About Equations
Author: Matt A. Bernstein
language: en
Publisher: John Wiley & Sons
Release Date: 2011-09-20
An accessible guide to developing intuition and skills for solving mathematical problems in the physical sciences and engineering Equations play a central role in problem solving across various fields of study. Understanding what an equation means is an essential step toward forming an effective strategy to solve it, and it also lays the foundation for a more successful and fulfilling work experience. Thinking About Equations provides an accessible guide to developing an intuitive understanding of mathematical methods and, at the same time, presents a number of practical mathematical tools for successfully solving problems that arise in engineering and the physical sciences. Equations form the basis for nearly all numerical solutions, and the authors illustrate how a firm understanding of problem solving can lead to improved strategies for computational approaches. Eight succinct chapters provide thorough topical coverage, including: Approximation and estimation Isolating important variables Generalization and special cases Dimensional analysis and scaling Pictorial methods and graphical solutions Symmetry to simplify equations Each chapter contains a general discussion that is integrated with worked-out problems from various fields of study, including physics, engineering, applied mathematics, and physical chemistry. These examples illustrate the mathematical concepts and techniques that are frequently encountered when solving problems. To accelerate learning, the worked example problems are grouped by the equation-related concepts that they illustrate as opposed to subfields within science and mathematics, as in conventional treatments. In addition, each problem is accompanied by a comprehensive solution, explanation, and commentary, and numerous exercises at the end of each chapter provide an opportunity to test comprehension. Requiring only a working knowledge of basic calculus and introductory physics, Thinking About Equations is an excellent supplement for courses in engineering and the physical sciences at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers, practitioners, and educators in all branches of engineering, physics, chemistry, biophysics, and other related fields who encounter mathematical problems in their day-to-day work.
Understanding Mathematical Concepts in Physics
Author: Sanjeev Dhurandhar
language: en
Publisher: Springer Nature
Release Date: 2024-07-16
Modern mathematics has become an essential part of today’s physicist’s arsenal and this book covers several relevant such topics. The primary aim of this book is to present key mathematical concepts in an intuitive way with the help of geometrical and numerical methods - understanding is the key. Not all differential equations can be solved with standard techniques. Examples illustrate how geometrical insights and numerical methods are useful in understanding differential equations in general but are indispensable when extracting relevant information from equations that do not yield to standard methods. Adopting a numerical approach to complex analysis it is shown that Cauchy’s theorem, the Cauchy integral formula, the residue theorem, etc. can be verified by performing hands-on computations with Python codes. Figures elucidate the concept of poles and essential singularities. Further the book covers topology, Hilbert spaces, Fourier transforms (discussing how fast Fourier transform works), modern differential geometry, Lie groups and Lie algebras, probability and useful probability distributions, and statistical detection of signals. Novel features include: (i) Topology is introduced via the notion of continuity on the real line which then naturally leads to topological spaces. (ii) Data analysis in a differential geometric framework and a general description of χ2 discriminators in terms of vector bundles. This book is targeted at physics graduate students and at theoretical (and possibly experimental) physicists. Apart from research students, this book is also useful to active physicists in their research and teaching.
Understanding Engineering Mathematics
Studying engineering, whether it is mechanical, electrical or civil relies heavily on an understanding of mathematics. This new textbook clearly demonstrates the relevance of mathematical principles and shows how to apply them to solve real-life engineering problems. It deliberately starts at an elementary level so that students who are starting from a low knowledge base will be able to quickly get up to the level required. Students who have not studied mathematics for some time will find this an excellent refresher. Each chapter starts with the basics before gently increasing in complexity. A full outline of essential definitions, formulae, laws and procedures are introduced before real world situations, practicals and problem solving demonstrate how the theory is applied. Focusing on learning through practice, it contains examples, supported by 1,600 worked problems and 3,000 further problems contained within exercises throughout the text. In addition, 34 revision tests are included at regular intervals. An interactive companion website is also provided containing 2,750 further problems with worked solutions and instructor materials