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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 62,645 ratings · 1,836 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 a writer and avid reader based in the Pacific Northwest. Before pursuing writing full-time, they spent over a decade working in education. When not working on their next book, Wolfgang Kühnel can usually be found hiking with their dog or trying out new coffee shops in their neighborhood.

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Andrew Rodriguez May 25, 2025
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Mark Mitchell August 11, 2024
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Laura Collins September 02, 2024
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Sharon Phillips February 13, 2025
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Eric Thompson December 06, 2024
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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Nicholas Smith
Nicholas Smith October 03, 2024
A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars.
John Green
John Green August 29, 2024
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David Nelson November 30, 2025
A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel.
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James Roberts April 28, 2025
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Cynthia Allen
Cynthia Allen February 05, 2024
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.
Melissa Sanchez
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Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
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Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
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Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem.
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A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece.
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Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre.
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I was completely blown away. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author.
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Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
Kenneth Collins
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A profoundly satisfying read. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Melissa Allen
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Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend.
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Karen Roberts
Karen Roberts June 10, 2025
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I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre.
John Clark
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Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
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Masterpiece from start to finish. 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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Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
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Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
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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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I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. A solid 4 stars from me.
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Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. 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. Worth every moment spent reading.
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Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading.
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Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre.
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I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
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Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
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Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre.
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Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem.
Rebecca Martin
Rebecca Martin December 09, 2024
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.
Elizabeth Wright
Elizabeth Wright January 15, 2024
Such a rich narrative. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece.
Brian Wilson
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A decent, if standard, experience. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Perfectly average.
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I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
Jessica Robinson
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I thoroughly enjoyed the elegance of it. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. Quite impressive overall.
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A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre.
James Nelson
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A highly sophisticated read. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall.
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Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author.
Matthew Flores
Matthew Flores January 13, 2025
Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
Joshua Nguyen
Joshua Nguyen September 28, 2024
Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall.
Melissa Adams
Melissa Adams May 28, 2024
It was adequate. Certain acts were fascinating, while others felt like a slight endeavor to complete. Perfectly average.
Ryan Reyes
Ryan Reyes July 27, 2025
I thoroughly enjoyed the elegance of it. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A wonderful companion for a quiet weekend.
Mary Collins
Mary Collins August 10, 2024
I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel.
Nicholas Hill
Nicholas Hill May 28, 2025
A highly sophisticated read. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel.
Karen Collins
Karen Collins October 17, 2025
I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. I would highly recommend this to discerning readers.
Ashley Edwards
Ashley Edwards May 11, 2024
I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre.
William Campbell
William Campbell April 16, 2024
Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
David Garcia
David Garcia April 07, 2025
A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel.
Karen Mitchell
Karen Mitchell June 17, 2024
It was adequate. It verged on repetitive in the second act. A more concise edit would have served it well. Unlikely to reread, but a reasonable expenditure of time.
Christopher Hill
Christopher Hill November 18, 2024
Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Elizabeth Williams
Elizabeth Williams June 09, 2025
A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Kenneth Jones
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Took a moment to build, but the payoff was spectacular. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel.
Elizabeth Williams
Elizabeth Williams June 07, 2025
Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading.
Donald  Parker
Donald Parker April 07, 2024
Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading.
Anthony Flores
Anthony Flores November 07, 2025
Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre.
David Cruz
David Cruz May 30, 2025
Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre.
Betty Gonzalez
Betty Gonzalez October 29, 2025
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.
David Adams
David Adams January 09, 2025
Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An effortless 5 stars.
Michael  Parker
Michael Parker October 31, 2025
Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading.
Eric Turner
Eric Turner September 27, 2025
Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars.
Mary Lee
Mary Lee January 05, 2025
Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Christopher Hill
Christopher Hill December 24, 2025
Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
Edward Torres
Edward Torres September 30, 2025
I thoroughly enjoyed the elegance of it. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall.
Rebecca Campbell
Rebecca Campbell February 07, 2025
Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
John Green
John Green August 29, 2024
Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre.
Jacob White
Jacob White January 09, 2025
Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading.
William  Davis
William Davis July 04, 2024
I was completely blown away. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Rebecca Thomas
Rebecca Thomas September 04, 2025
Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars.
Paul Hill
Paul Hill February 11, 2024
A very strong entry. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers.
Jennifer Walker
Jennifer Walker November 02, 2024
I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. A solid 4 stars from me.
Donna Lewis
Donna Lewis March 13, 2025
Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem.
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