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The second volume of this introduction into analysis deals with the integration theory of functions of one variable, the multidimensional differential calculus and the theory of curves and line integrals. The modern and clear development that started in Volume I (3-7643-7153-6) continues. In this way a sustainable basis will be created which allows to deal with interesting applications that sometimes go considerably beyond the material that is represented in traditional textbooks. This applies, for instance, to the exploration of Nemytskii operators which enable a transparent introduction into the calculus of variations and the derivation of the Euler-Lagrange equations. Another example is the presentation of the local theory of submanifolds of Rn.
As with the first, the second volume contains substantially more material than can be covered in a one-semester course. Such courses may omit many beautiful and well-grounded applications which connect broadly to many areas of mathematics. We of course hope that students will pursue this material independently; teachers may find it useful for undergraduate seminars. For an overview of the material presented, consult the table of contents and the chapter introductions. As before, we stress that doing the numerous exercises is indispensable for understanding the subject matter, and they also round out and amplify the main text. In writing this volume, we are indebted to the help of many. We especially thank our friends and colleagues Pavol Quittner and Gieri Simonett. They have not only meticulously reviewed the entire manuscript and assisted in weeding out errors but also, through their valuable suggestions for improvement, contributed essentially to the final version. We also extend great thanks to our staff for their careful perusal of the entire manuscript and for tracking errata and inaccuracies. Our most heartfelt thank extends again to our "typesetting perfectionist", 1 without whose tireless effort this book would not look nearly so nice. We also thank Andreas for helping resolve hardware and software problems. Finally, we extend thanks to Thomas Hintermann and to Birkhauser for the good working relationship and their understanding of our desired deadlines.
As with the first, the second volume contains substantially more material than can be covered in a one-semester course. Such courses may omit many beautiful and well-grounded applications which connect broadly to many areas of mathematics. We of course hope that students will pursue this material independently; teachers may find it useful for undergraduate seminars. For an overview of the material presented, consult the table of contents and the chapter introductions. As before, we stress that doing the numerous exercises is indispensable for understanding the subject matter, and they also round out and amplify the main text. In writing this volume, we are indebted to the help of many. We especially thank our friends and colleagues Pavol Quittner and Gieri Simonett. They have not only meticulously reviewed the entire manuscript and assisted in weeding out errors but also, through their valuable suggestions for improvement, contributed essentially to the final version. We also extend great thanks to our staff for their careful perusal of the entire manuscript and for tracking errata and inaccuracies. Our most heartfelt thank extends again to our "typesetting perfectionist", 1 without whose tireless effort this book would not look nearly so nice. We also thank Andreas for helping resolve hardware and software problems. Finally, we extend thanks to Thomas Hintermann and to Birkhauser for the good working relationship and their understanding of our desired deadlines.