Many-Minds Relativity

iii Preface Applied Mathematics: Body&Soul is a mathematics education reform program including a series of books, together with associated educational material and open source software freely available from the project web page at www.bodysoulmath.org. Body&Soul reflects the revolutionary new possibilities of mathematical modeling opened by the modern computer in the form of Computational Calculus (CC), which is now changing the paradigm of mathematical mod-eling in science and technology with new methods, questions and answers, as a modern form of the classical calculus of Leibniz and Newton. The Body&Soul series of books presents CC in a synthesis of computational mathematics (Body) and analytical mathematics (Soul) including applications. Volumes 1-3 [41] give a modern version of calculus and linear algebra including computation starting at a basic undergraduate level, and subsequent volumes on a graduate level cover different areas of applications with focus mainly on computational methods: The present book is Volume 5 and is a digression into (and out of) Einstein's special theory of relativity. A gentle introduction to the Body&Soul series is given in [54]. The overall goal of the Body&Soul project may be formulated as the Automation of Computational Mathematical Modeling (ACMM) involving the key steps of automation of (i) discretization, (ii) optimization and (iii) modeling. The objective of the ACMM is to open for massive use of CC in science, engineering, medicine, and other areas of application. ACMM is realized in the FEniCS project (www.fenics.org), which represents the top software part of Body&Soul. The automation of discretization (i) involves automatic translation of a given differential equation in standard mathematical notation into a discrete system of equations, which can be automatically solved using numerical linear algebra to produce an approximate solution of the differential equation. The translation is performed using adaptive stabilized finite element methods, which we refer to as General Galerkin or G2 with the iv adaptivity based on a posteriori error estimation by duality and the stabilization representing a weighted least squares control of the residual. The automation of optimization (ii) is performed similarly starting from the differential equations expressing stationarity of an associated Lagrangian. Finally, one can couple modeling to optimization by seeking from an Ansatz a model with best fit to given data. Vol 4 [55] presents a test of the functionality of the general technique for ACMM based on G2 applied to the specific problem of solving the incompressible Euler and Navier–Stokes (NS) equations computationally. …