## Multidimensional Real Analysis I: DifferentiationPart one of the authors' comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of differential analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains many results and exercises that cannot be found elsewhere. The numerous exercises illustrate a variety of applications in mathematics and physics. This combined with the exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study. |

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### Contents

1 | |

Chapter 2 Differentiation | 37 |

Chapter 3 Inverse Function and Implicit Function Theorems | 87 |

Chapter 4 Manifolds | 107 |

Chapter 5 Tangent Spaces | 133 |

### Common terms and phrases

apply assertion assume Aut(R bijective Cauchy sequence chain rule Ck diffeomorphism Ck mapping coefﬁcients compact Conclude consider convergent coordinates curve Deduce define Deﬁnition denote diffeomorphism differentiable End(R equals equation Example exist an open ﬁnd ﬁrst formula function f Furthermore G R2 geometric given grad Hint identity Illustration for Exercise Implicit Function Theorem implies inequality injective integration intersection Inverse Function Theorem isomorphism Lemma Let f Lie algebra Lin(R Lin(Rn,Rp linear space linear subspace Lipschitz continuous manifold mapping f Mat(n Mat(n,R mathematical induction matrix needed for Exercises notation obtain open neighborhood open set open subset orthogonal parametrization partial derivatives polynomial Proof Prove that f R2 of dimension respectively rotation satisﬁes satisfying sequel to Exercise sequence Show submanifold in R2 Suppose surjective tangent space tangent vector variables Verify