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Algebraic curves and Riemann surfaces for undergraduates = the theory of the donut /

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Algebraic curves and Riemann surfaces for undergraduates/ by Anil Nerode, Noam Greenberg.
其他題名:	the theory of the donut /
作者:	Nerode, Anil.
其他作者:	Greenberg, Noam.
出版者:	Cham :Springer International Publishing : : 2022.,
面頁冊數:	xvi, 450 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
標題:	Global Analysis and Analysis on Manifolds. -
電子資源:	https://doi.org/10.1007/978-3-031-11616-2
ISBN:	9783031116162

Algebraic curves and Riemann surfaces for undergraduates = the theory of the donut /
Nerode, Anil.

Algebraic curves and Riemann surfaces for undergraduatesthe theory of the donut /[electronic resource] :by Anil Nerode, Noam Greenberg. - Cham :Springer International Publishing :2022. - xvi, 450 p. :ill., digital ;24 cm.

1 Introduction -- Part I Algebraic curves -- 2 Algebra -- 3 Affine space -- 4 Projective space -- 5 Tangents -- 6 Bézout's theorem -- 7 The elliptic group -- Part II Riemann Surfaces -- 8 Quasi-Euclidean spaces -- 9 Connectedness, smooth and simple -- 10 Path integrals -- 11 Complex differentiation -- 12 Riemann surfaces -- Part III Curves and surfaces -- 13 Curves are surfaces -- 14 Elliptic functions and the isomorphism theorem -- 15 Puiseux theory -- 16 A brief history of elliptic functions.

The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or "donut") is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric "chord-and-tangent" method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses.

ISBN: 9783031116162

Standard No.: 10.1007/978-3-031-11616-2doiSubjects--Topical Terms:

672519
Global Analysis and Analysis on Manifolds.

LC Class. No.: QA565 / .N47 2022

Dewey Class. No.: 516.352

Algebraic curves and Riemann surfaces for undergraduates = the theory of the donut /
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505 0 $a 1 Introduction -- Part I Algebraic curves -- 2 Algebra -- 3 Affine space -- 4 Projective space -- 5 Tangents -- 6 Bézout's theorem -- 7 The elliptic group -- Part II Riemann Surfaces -- 8 Quasi-Euclidean spaces -- 9 Connectedness, smooth and simple -- 10 Path integrals -- 11 Complex differentiation -- 12 Riemann surfaces -- Part III Curves and surfaces -- 13 Curves are surfaces -- 14 Elliptic functions and the isomorphism theorem -- 15 Puiseux theory -- 16 A brief history of elliptic functions.
520 $a The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or "donut") is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric "chord-and-tangent" method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses.
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