04019nam a22005295i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001700153072001600170072002300186072001500209100008200224245015400306250001800460264006900478300006500547336002600612337002600638338003600664347002400700490004900724505046300773520166101236650002602897650002702923650002202950650002402972650001602996650002803012650005703040650002603097650001403123650002203137650003903159650002803198710003403226773002603260776003603286776003603322776003603358830004903394856004603443978-3-031-60394-5DE-He21320251030112632.0cr nn 008mamaa240716s2024    sz |    s    |||| 0|eng d  a97830316039457 a10.1007/978-3-031-60394-52doi 4aQC19.2-20.85 7aPHU2bicssc 7aSCI0400002bisacsh 7aPHU2thema1 aDhurandhar, Sanjeev.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aUnderstanding Mathematical Concepts in Physicsh[electronic resource] :bInsights from Geometrical and Numerical Approaches /cby Sanjeev Dhurandhar.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXVI, 351 p. 57 illus., 42 illus. in color.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aLecture Notes in Physics,x1616-6361 ;v10300 aDedication -- 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.  aModern 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. 0aMathematical physics. 0aMathematical analysis. 0aFourier analysis. 0aTopological groups. 0aLie groups. 0aDifferential equations.14aTheoretical, Mathematical and Computational Physics.24aMathematical Physics.24aAnalysis.24aFourier Analysis.24aTopological Groups and Lie Groups.24aDifferential Equations.2 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303160393808iPrinted edition:z978303160395208iPrinted edition:z9783031603969 0aLecture Notes in Physics,x1616-6361 ;v103040uhttps://doi.org/10.1007/978-3-031-60394-5