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Differential Geometry: Curves - Surfaces - Manifolds (Student Mathematical Library, Volume 16)

Differential Geometry: Curves - Surfaces - Manifolds

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Differential Geometry: Curves - Surfaces - Manifolds (Student Mathematical Library, Volume 16)

Differential Geometry: Curves - Surfaces - Manifolds

4.85 22,615 ratings · 6,274 reviews
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Overview: Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to helpalong the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.

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Pages N/A
Language Eng
Published January 1, 2002
ISBN 1567302
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Wolfgang Kühnel is the author of several novels. Originally from the East Coast, they now live in the Midwest with family. In addition to writing, Wolfgang Kühnel enjoys gardening, cooking, and reading way past bedtime. Their work often explores the complexities of everyday life and human relationships.

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Andrew Rodriguez
Andrew Rodriguez December 12, 2024
Exceeded all my expectations. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars.
Carol Taylor
Carol Taylor September 20, 2024
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Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
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Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem.
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A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading.
Edward Allen
Edward Allen June 24, 2024
Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem.
Brian Jackson
Brian Jackson March 03, 2024
Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre.
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Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall.
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Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
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Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
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Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem.
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Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
Lisa Ramirez
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I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. A wonderful companion for a quiet weekend.
Matthew Adams
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Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author.
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A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
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Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading.
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Such a rich narrative. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I am now a lifelong fan of this author.
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Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Elizabeth Allen
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A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author.
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Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An effortless 5 stars.
Brian Walker
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A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel.
Lisa Martin
Lisa Martin April 01, 2025
Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Jacob Campbell
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Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author.
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Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
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I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. An effortless 5 stars.
Susan Jackson
Susan Jackson June 06, 2024
I thoroughly enjoyed the elegance of it. The protagonist was compelling, though some subplots felt slightly overshadowed. I would highly recommend this to discerning readers.
Thomas Torres
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A highly sophisticated read. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel.
Matthew Moore
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Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Kimberly White
Kimberly White March 05, 2024
A decent, if standard, experience. It verged on repetitive in the second act. A more concise edit would have served it well. A fair 3 stars.
Joshua Walker
Joshua Walker November 19, 2024
I find myself somewhat conflicted. It verged on repetitive in the second act. A more concise edit would have served it well. Perfectly average.
Melissa Phillips
Melissa Phillips July 18, 2024
A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem.
Margaret Evans
Margaret Evans August 18, 2024
A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend.
Daniel Rodriguez
Daniel Rodriguez October 15, 2025
A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre.
Michelle Rodriguez
Michelle Rodriguez October 28, 2024
A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Ryan Martin
Ryan Martin December 02, 2024
A very strong entry. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel.
Daniel Hall
Daniel Hall August 19, 2025
Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
Melissa Evans
Melissa Evans March 18, 2025
I find myself somewhat conflicted. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Perhaps more aligned with a different audience's tastes.
Kenneth Baker
Kenneth Baker December 23, 2025
Took a moment to build, but the payoff was spectacular. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. Quite impressive overall.
Mary Phillips
Mary Phillips November 07, 2025
Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre.
Steven Ramirez
Steven Ramirez October 31, 2024
An acceptable diversion. The prose was competent, yet the characters lacked the depth I usually seek. A fair 3 stars.
Christopher Nguyen
Christopher Nguyen March 24, 2025
Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars.
Matthew Walker
Matthew Walker August 18, 2025
Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem.
Margaret Hernandez
Margaret Hernandez November 30, 2024
A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. Quite impressive overall.
Carol Perez
Carol Perez December 18, 2024
A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A very distinguished novel.
Ashley  Davis
Ashley Davis April 06, 2024
I thoroughly enjoyed the elegance of it. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall.
Barbara Reyes
Barbara Reyes September 21, 2024
A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A very distinguished novel.
Cynthia Rodriguez
Cynthia Rodriguez May 06, 2025
Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Emily Green
Emily Green July 31, 2025
A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Thomas Torres
Thomas Torres January 04, 2025
A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
Amanda Flores
Amanda Flores April 26, 2025
Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author.
Michelle Wilson
Michelle Wilson October 14, 2025
Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Michelle Nelson
Michelle Nelson May 09, 2025
A profoundly satisfying read. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem.
Kenneth Perez
Kenneth Perez August 07, 2024
A very strong entry. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. Quite impressive overall.
Kimberly White
Kimberly White December 17, 2024
Excellent, though slightly flawed. It held my rapt attention, even when certain twists were telegraphed. A very distinguished novel.
Dorothy Wilson
Dorothy Wilson August 20, 2024
Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author.
Mark Lewis
Mark Lewis September 01, 2024
Excellent, though slightly flawed. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend.
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