{"title":"Index","authors":"","doi":"10.1017/9781108557917.036","DOIUrl":"https://doi.org/10.1017/9781108557917.036","url":null,"abstract":"","PeriodicalId":418951,"journal":{"name":"Mathematics for Physicists","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122620566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Calculus of complex functions","authors":"A. Altland, J. Delft","doi":"10.1017/9781108557917.022","DOIUrl":"https://doi.org/10.1017/9781108557917.022","url":null,"abstract":"The book introduces complex analysis as a natural extension of the calculus of real-valued functions. The mechanism for doing so is the extension theorem , which states that any real analytic function extends to an analytic function defined in a region of the complex plane. The connection to real functions and calculus is then natural. The introduction to analytic functions feels intuitive and their fundamental properties are covered quickly. As a result, the book allows a surprisingly large cover-age of the classical analysis topics of analytic and meromorphic functions, harmonic functions, contour integrals and series representations, conformal maps, and the Dirichlet problem.","PeriodicalId":418951,"journal":{"name":"Mathematics for Physicists","volume":"20 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113943013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Curvilinear coordinates","authors":"N. Deruelle, J. Uzan","doi":"10.1017/cbo9781139012973.022","DOIUrl":"https://doi.org/10.1017/cbo9781139012973.022","url":null,"abstract":"This chapter presents a discussion on curvilinear coordinates in line with the introduction on Cartesian coordinates in Chapter 1. First, the chapter introduces a new system C of curvilinear coordinates xⁱ = xⁱ(Xj) (also sometimes referred to as Gaussian coordinates), which are nonlinearly related to Cartesian coordinates. It then introduces the components of the covariant derivative, before considering parallel transport in a system of curvilinear coordinates. Next, the chapter shows how connection coefficients of the covariant derivative as well as the Euclidean metric can be related to each other. Finally, this chapter turns to the kinematics of a point particle as well as the divergence and Laplacian of a vector and the Levi-Civita symbol and the volume element.","PeriodicalId":418951,"journal":{"name":"Mathematics for Physicists","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121089819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Problems: Linear Algebra","authors":"H. Pétard","doi":"10.1017/9781108557917.012","DOIUrl":"https://doi.org/10.1017/9781108557917.012","url":null,"abstract":"In the following R denotes the field of real numbers while C denotes the field of complex numbers. In general, U, V , and W denote vector spaces. The set of all linear transformations from V into W is denoted by L(V, W) , while L(V) denotes the set of linear operators on V. For a linear transformation T , the null space of T (also known as the kernel of T) is denoted by null T , while the range space of T (also known as the image of T), is denoted by range T. Problem 0. Let V be a finite-dimensional vector space and let T be a linear operator on V. Suppose that T commutes with every diagonalizable linear operator on V. Prove that T is a scalar multiple of the identity operator. Problem 1. Let V and W be vector spaces and let T be a linear transformation from V into W. Suppose that V is finite-dimensional. Prove rank(T) + nullity(T) = dim V. Problem 2. Let A and B be n × n matrices over a field F (1) Prove that if A or B is nonsingular, then AB is similar to BA. (2) Show that there exist matrices A and B so that AB is not similar to BA. (3) What can you deduce about the eigenvalues of AB and BA. Prove your answer.","PeriodicalId":418951,"journal":{"name":"Mathematics for Physicists","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129321945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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