## Analysis in Vector SpacesA rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes: -
Sets and functions -
Real numbers -
Vector functions -
Normed vector spaces -
First- and higher-order derivatives -
Diffeomorphisms and manifolds -
Multiple integrals -
Integration on manifolds -
Stokes' theorem -
Basic point set topology
Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants. Assuming only a fundamental understanding of linear algebra and single variable calculus, |

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### Contents

Sets and Functions | 3 |

Real Numbers | 31 |

Vector Functions | 61 |

Normed Vector Spaces | 123 |

Derivatives | 175 |

Diffeomorphisms and Manifolds | 225 |

HigherOrder Derivatives | 267 |

Multiple Integrals | 287 |

Integration on Manifolds | 339 |

StokesTheorem | 381 |

Construction of the real numbers | 419 |

Dimension of a vector space | 431 |

Partitions of unity | 451 |

### Other editions - View all

Analysis in Vector Spaces, Solutions Manual Mustafa A. Akcoglu,Paul F.A. Bartha,Dzung Minh Ha No preview available - 2009 |

Analysis in Vector Spaces Set Mustafa A. Akcoglu,Paul F. A. Bartha,Dzung Minh Ha No preview available - 2009 |