Differential Geometry (Wiley Classics Library) 1st Edition by J. J. Stoker (PDF)

Description

This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.

User’s Reviews

Editorial Reviews: About the Author James J Stoker was an American applied mathematician and engineer. He was director of the Courant Institute of Mathematical Sciences and is considered one of the founders of the institute, Courant and Friedrichs being the others. Stoker is known for his work in differential geometry and theory of water waves.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This was the classic grad school introduction to differential geometry before Do Carmo. I guess it was tried as an undergraduate text but was also dropped in favor of Millman and Parker or Do Carmo. It’s clearly advanced undergrad or beginning grad level. It requires the same background namely mathematical analysis and linear algebra but requires students to fill in gaps on their own. For example the proof of Synge’s Theorem requires homotopy and the author gives an indication of what it is with a reference for details whereas Do Carmo does give a full explanation. This may seem like a negative but this is rare. So Do Carmo is more basic which is probably why it superseded this text. But part of the reason this occurred also was the greater number of problems in Do Carmo. A plus? Not so fast. Many of the main proofs in Do Carmo rely on the student to fill in proof of critical steps as problems and this was sometimes at a higher level of difficulty than the material presented. Let’s say without an engaged instructor as is too often the case you won’t get your money’s worth. Stoker gives complete proofs of the main results and reserves the problems for clarifying proof ideas or extending results.About the first half of this text covers what you’ll see in Do Carmo.After that he does the intrinsic differential geometry of n-dimensional manifolds using the tensor methods of Cartan and Weyl. You’ll find here that parallel translation, covariant differentiation and the Riemann Curvature Tensor all rely on operating on two dimensional as in the first part, a point not easily seen in Do Carmo This culminates in a brief introduction to general relativity taking you as far as showing that the metric coefficients in the Riemann metric in a free-fall frame look to be approximately the Newtonian gravitational potentials! The final chapter deals with Cartan frames and the Structural Equations, so differential forms are needed. Some results particularly in the final chapter are cited in Do Carmo since he doesn’t cover these more advanced methods.I’ll give some details as to Stoker’s viewpoint in this text. J.J.Stoker was majoring in applied mathematics and mechanics until his differential geometry course with Heinz Hopf-he took his thesis in differential geometry. Though a differential geometer most of his work at the Courant Institute was in mathematical physics and mechanics. There’s significant use of the calculus of variations particularly in regard to geodesics later in the text. He expects you’ve heard of the Euler-Lagrange equations as you’ll use them in a problem to show that the geodetic equations result when these are applied to the functional integral which is path length using a Riemann metric. The Riemann metric is really a generalized kinetic energy with time parameter when looked at as a lagrangian here. Differential geometry with calculus of variations is the subject of Morse Theory and the classic reference is Milnor. Anyway Stoker’s broad background will show up in enlightening comments or problems. Again regarding geodesics, Stoker emphasizes Gauss’s intrinsic approach to differential geometry,i.e., independent of embedding. Clearly the Euler-Lagrange approach just mentioned is intrinsic-the results only depend on the metric coefficients. In the earlier part of the text a geodesic is defined as a curve of geodetic curvature zero and the same geodetic equations as found with Euler-Lagrange result but this is only after he demonstrates that geodetic curvature is intrinsic. In some introductory texts geodesics are defined as curves with acceleration normal to the surface and without concern about the embedding, not nice. I’ll give an example which the engaged reader might try-Take the cylindrical polar coordinate metric in 3 space as usual, take r constant so you’re on the cylinder surface, now use the geodetic equations you found with Euler-Lagrange on this metric-you’ll find you get a helix or helices for geodesics.In conclusion, if you could only have just one text to tell you what differential geometry is about in your library-This Is It.

⭐A classic

⭐Most books in the Wiley Classics collection are not for beginners in the subject. This is one of the few that you can use as either an introduction to differential geometry or as a reference. Another flaw found in most mathematical texts is the order in which ideas are presented. There are so many areas of mathematics related to so many other areas that it can be very difficult for a professor or author to present a linear course of thought on the subject, Stoker does an excellent job of finding that linear course of thought. I highly recommend it; even if you have to pay full price for it.Just so you know Stoker’s Differential Geometry is undergraduate level differential geometry.The 2 volume set by Shoshichi Kobayashi and Katsumi Nomizu in the Wiley Classics Library is a graduate level treatment of the subject.

⭐The book is very accurate and detailed.I would only like to add to the other reviews that other must in the subject are Schaum’s Outline of Differential Geometry and Struik.

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