Learning and teaching mathematics using simulations : plus 2000 examples from physics / Dieter Röss.

This is a unique, comprehensive and documented collection of simulations in mathematics and physics: More than 2000 simulations, offered on our webpage for comfortable use online. The book, written by an experienced teacher and practitioner, contains a complete introduction to mathematics and the documentation to the simulations. This is a great way to learn mathematics and physics. Suitable for courses in Mathematics for Engineering and Sciences.

Introduction; Goal and structure of the digital book; Directories; Usage and technical conventions; Example of a simulation: The Möbius band; Physics and mathematics; Mathematics as the ``Language of physics''; Physics and calculus; Numbers; Natural numbers; Whole numbers; Rational numbers; Irrational numbers; Algebraic numbers; Transcendental numbers; and the quadrature of the circle, according to Archimedes; Real numbers; Complex numbers; Representation as a pair of real numbers; Normal representation with the ``imaginary unit i''; Complex plane; Representation in polar coordinates.

Simulation of complex addition and subtractionSimulation of complex multiplication and division; Extension of arithmetic; Sequences of numbers and series; Sequences and series; Sequence and series of the natural numbers; Geometric series; Limits; Fibonacci sequence; Complex sequences and series; Complex geometric sequence and series; Complex exponential sequence and exponential series; Influence of limited accuracy of measurements and nonlinearity; Numbers in mathematics and physics; Real sequence with nonlinear creation law: Logistic sequence.

Complex sequence with nonlinear creation law: FractalsFunctions and their infinitesimal properties; Definition of functions; Difference quotient and differential quotient; Derivatives of a few fundamental functions; Powers and polynomials; Exponential function; Trigonometric functions; Rules for the differentiation of combined functions; Derivatives of further fundamental functions; Series expansion: the Taylor series; Coefficients of the Taylor series; Approximation formulas for simple functions; Derivation of formulas and errors bounds for numericaldifferentiation.

Interactive visualization of Taylor expansionsGraphical presentation of functions; Functions of one to three variables; Functions of four variables: World line in the theory of relativity; General properties of functions y=f(x); Exotic functions; The limiting process for obtaining the differential quotient; Derivatives and differential equations; Phase space diagrams; Antiderivatives; Definition of the antiderivative via its differential equation; Definite integral and initial value; Integral as limit of a sum; The definition of the Riemann integral; Lebesgue integral.

Rules for the analytical integrationNumerical integration methods; Error estimates for numerical integration; Series expansion (2): the Fourier series; Taylor series and Fourier series; Determination of the Fourier coefficients; Visualizing the calculation of coefficients and spectrum; Examples of Fourier expansions; Complex Fourier series; Numerical solution of equations and iterative methods; Visualization of functions in the space of real numbers; Standard functions y=f(x); Some functions y=f(x) that are important in physics; Standard functions of two variables z=f(x, y); Waves in space.

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