## Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda CalculusThis second volume of a comprehensive tour through mathematical core subjects for computer scientists completes the ?rst volume in two - gards: Part III ?rst adds topology, di?erential, and integral calculus to the t- ics of sets, graphs, algebra, formal logic, machines, and linear geometry, of volume 1. With this spectrum of fundamentals in mathematical e- cation, young professionals should be able to successfully attack more involved subjects, which may be relevant to the computational sciences. In a second regard, the end of part III and part IV add a selection of more advanced topics. In view of the overwhelming variety of mathematical approaches in the computational sciences, any selection, even the most empirical, requires a methodological justi?cation. Our primary criterion has been the search for harmonization and optimization of thematic - versity and logical coherence. This is why we have, for instance, bundled such seemingly distant subjects as recursive constructions, ordinary d- ferential equations, and fractals under the unifying perspective of c- traction theory. |

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### Contents

Topology and Calculus | 3 |

Differentiability | 37 |

Inverse and Implicit Functions 59 | 58 |

Integration | 73 |

The Fundamental Theorem of Calculus and Fubinis Theorem | 87 |

Vector Fields 97 | 96 |

Fixpoints | 105 |

Main Theorem of ODEs | 113 |

Splines 161 | 160 |

Fourier Theory | 183 |

Wavelets | 215 |

Fractals 231 | 230 |

Neural Networks | 253 |

Probability Theory 279 | 278 |

Lambda Calculus | 313 |

A Further Reading 335 | 334 |

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