Understanding Mathematical Concepts in Physics

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.

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.