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56 of 57 people found the following review helpful:
4.0 out of 5 stars
Differential Geometry - A Schaum's Outline Series, June 25, 2000
This review is from: Schaum's Outline of Differential Geometry (Schaum's) (Paperback)
As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject.
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19 of 19 people found the following review helpful:
4.0 out of 5 stars
Good as a basic textbook and a source of solve problems, June 4, 2006
This review is from: Schaum's Outline of Differential Geometry (Schaum's) (Paperback)
This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus. Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.
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6 of 6 people found the following review helpful:
5.0 out of 5 stars
A practical elementary introduction to classical differential geometry, May 30, 2009
This review is from: Schaum's Outline of Differential Geometry (Schaum's) (Paperback)
After so many years, this book continues to be a valuable introduction to the differential geometry (DG) of curves and surfaces in the euclidean 3-dimensional space R^3, quite clear and efficient for self study, since each chapter combines a serious bulk of theory and many solved exercises, as well as some unsolved problems. It starts reviewing much of the necessary calculus needed. Then, it goes into curves, defining curvature and torsion, and proving the Frenet-Serret equations. It is shown that every regular curve is detrmined by its curvature and torsion (up to a rigid motion). Many interesting problems on curves illustrate the theory. But little attention is given to plane curves and no global property of curves is given (what is quite understandable, since they are hard to prove). The book continues with surfaces, defining parametrizations, atlas, the tangent plane and the differential of a map of surfaces. Then, we find an excellent introductory exposition of lines of curvature and assymptotic lines, (including Meusnier, Euler, Rodrigues and Beltrami-Enneper theorems) as well as geodesic curvature, geodesics, mean and Gauss curvature. The so called fundamental existence and unicity theorems for curves and surfaces in R^3 are stated and proved, as well as Gauss Theorema Egregium. However, there is no mention of parallel transport (you can find this in Stoker Differential Geometry (Wiley Classics Library), in Goetz Introduction to Differential Geometry (Addison-Wesley Series in Mathematics), in Millman-Parker Elements of Differential Geometry's, in do Carmo's Differential Geometry of Curves and Surfaces or in Klingenberg's A Course in Differential Geometry (Graduate Texts in Mathematics), all of them introductory books on DG too. The book also treats the simplest global properties of surfaces: (1) orientability (mildly presented), (2) Liebmann's theorem characterising compact connected surfaces of constant curvature in R^3 as spheres ( clearly proved, without assuming its orientabilty), (3) Gauss-Bonnet theorem, proved in a rather sketchy way, but well illustrated in some exercises, which clarify its meaning and difficulty. In general, many theoretical properties are proved as exercises. Practical questions are easy or not too hard to solve. If you really don't know the subject, this book is a perfect start, alone or combined with those previously cited works, or with Struik's classical Lectures on Classical Differential Geometry: Second Edition, Oprea's Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks), or Montiel-Ros' Curves and Surfaces (Graduate Studies in Mathematics). Other problem books on the DG of curves and surfaces are rare. I will mention (1) Fedenko's (Mir Editions, now re-edited by USSR (sic!) editions-Moscow) (similar to M. Lipschutz's level, but much less detailed and with no theory). (2) Mishchenko-Solovyev-Fomenko (Problems in DG and Topology, Mir- Moscow).
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