## Applied Mathematics: Body and Soul: Volume 1: Derivatives and Geometry in IR3Applied Mathematics: Body & Soul is a mathematics education reform project developed at Chalmers University of Technology and includes a series of volumes and software. The program is motivated by the computer revolution opening new possibilities of computational mathematical modeling in mathematics, science and engineering. It consists of a synthesis of Mathematical Analysis (Soul), Numerical Computation (Body) and Application. Volumes I-III present a modern version of Calculus and Linear Algebra, including constructive/numerical techniques and applications intended for undergraduate programs in engineering and science. Further volumes present topics such as Dynamical Systems, Fluid Dynamics, Solid Mechanics and Electro-Magnetics on an advanced undergraduate/graduate level. Volume I (Derivatives and Geometry in R3) presents basics of Calculus starting with the construction of the natural, rational, real and complex numbers, and proceeding to analytic geometry in two and three space dimensions, Lipschitz continuous functions and derivatives, together with a variety of applications. Volume II (Integrals and Geomtery in Rn) develops the Riemann integral as the solution to the problem of determining a function given its derivative, and proceeds to generalizations in the form of initial value problems for general systems of ordinary differential equations, including a variety of applications. Linear algebra including numerics is also presented. Volume III (Calculus in Several Dimensions) presents Calculus in several variables including partial derivatives, multi-dimensional integrals, partial differential equations and finite element methods, together with a variety of applications modeled as systems of partial differential equations. The authors are leading researchers in Computational Mathematics who have written various successful books. Further information on Applied Mathematics: Body and Soul can be found at http://www.phi.chalmers.se/bodysoul/. |

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### Contents

Derivatives and Geometry in R | 2 |

Solving Linear Algebraic Systems 651 | 3 |

Linear Algebra Tool Bag 685 | 16 |

Optimization 871 | 18 |

Applications Tool Bag 1103 | 19 |

The Mathematics Laboratory | 21 |

Introduction to Modeling | 25 |

A Very Short Calculus Course | 33 |

Sequences and limits | 165 |

The Square Root of Two | 185 |

Real numbers | 195 |

The Bisection Algorithm for fa 0 | 215 |

Do Mathematicians Quarrel? | 221 |

The Function y a | 241 |

Analytic Geometry in R 265 | 264 |

Analytic Geometry in R | 313 |

Calculus Tool Bag I 585 | 34 |

The Integral 429 | 36 |

Natural Numbers and Integers | 47 |

Mathematical Induction 63 | 62 |

Rational Numbers | 71 |

Piecewise Linear Polynomials in R and | 76 |

Pythagoras and Euclid | 87 |

What is a Function? | 103 |

Polynomial functions | 119 |

Fourier Analysis Tool Bag 1185 | 120 |

Combinations of functions 141 | 140 |

Lipschitz Continuity | 149 |

FEM for Boundary Value Problems in R and R | 339 |

Complex Numbers | 345 |

Analytic Functions 1107 | 352 |

The Derivative 355 | 354 |

Differentiation Rules | 377 |

Newtons Method | 393 |

Galileo Newton Hooke Malthus and Fourier 403 | 402 |

Trigonometric Functions 505 | 409 |

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423 | |