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 2 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.

Aims at helping the reader understand the mathematical concepts required in physics Emphasises the understanding of differential equations and complex analysis through a numerical/geometrical approach Supports understanding through Python codes provided

Autorentext

Prof. Sanjeev Dhurandhar has published over hundred papers in top international journals such as Physical Reviews, MNRAS, Classical & Quantum Gravity, and also a review article (coauthored with M. Tinto) in Living Reviews published by Springer. About three quarters of the publications are on gravitational waves, mainly on their data analysis, and the rest are on various aspects of general relativity. Recently, he has published a book (Springer) entitled General Relativity and Gravitational Waves: Essentials of theory and Practice co-authored with Prof. Sanjit Mitra. The author led the gravitational wave group at IUCAA from 1989 to 2011 until he superannuated. He has taught the Mathematical Physics course at IUCAA at graduate level to physics students for decades and also in the mathematics & physics department of Pune University in the1980s. He has also given courses on differential geometry, probability theory etc. during his career at IUCAA. He holds a Master's degree in mathematics (he switched to physics for his Ph. D.). The author is a recipient of several prestigious awards which include the Vijnan Bhushan Firodia Award for Outstanding Contributions to Science, the Meghnad Saha Memorial Gold Medal (The Asiatic Society, Kolkata, India) for Outstanding Contributions to Physics, Milners breakthrough prize (Milners Foundation U.S.A.) awarded for the detection of gravitational waves (shared with the Ligo Science Collaboration), etc. He is a fellow of the American Physical Society (APS), the three prestigious Indian academies: (i) Indian National Science Academy, Delhi, (ii) the Indian Academy of Sciences, Bangalore, and (iii) National Academy of Sciences India, Allahabad.

Inhalt

Dedication.- Preface.- Topology.- Hilbert Spaces.- Fourier Analysis.- Complex analysis: hands on.- Understanding differential equations: geometrical insights and general analysis.- Solving Differential Equations.- Differential Geometry and Tensors.- Representations of the rotation group and Lie groups.- Probability and Random Variables.- Probability distributions in physics.- The statistical detection of signals in noisy data.- Bibliography.