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SOLID ANALYTIC GEOMETRY SOLID ANALYTIC GEOMETRY By ADRIAN ALBERT Professor of Mathematics The University of Chicago FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. NEW YORK TORONTO LONDON 1949 SOLID ANALYTIC GEOMETRY Copyright, 1949, by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. THE MAPLE PRESS COMPANY, YORK, PA. PREFACE In a recent text on college algebra the author gave a brief pre senlation of what seems to him to be the best basis for a modern course on plane analytic geometry, that is, the algebraic vector approach. The present text contains an extension of this approach, yielding an exposition in full for the three-dimensional case, and thereby ties up the study of space analytic geometry with the theory of vectors and matrices. Chapters 1 and 2 contain a treatment of the equations of lines and planes. After a preliminary study of the linear operations on n-dimensional vectors, rectangular coordinates are introduced, three-dimensional vectors and inner products are interpreted geometrically, and, from a consideration of scalar products and axis translations, the parametric equations of a line are obtained. The vector approach then yields a very simple derivation of the normal form of an equation of a plane, and the standard forms of plane and line equations are rather immediate consequences. Chapter 3 contains an exposition of classical elementary sur face and curve theory. Chapter 4 contains the usual treatment of spheres, and Chapter 5 gives the classical descriptions of quadric surfaces in standard position. The latter chapter ends with a rather novel classification of quadrics according to certain invariants. Chapter 6 is an exposition of that part of the theory of matrices which is needed for a complete development of the so-called principal axis transformation. A full account of the orthogonal reduction of a real quadratic form in n variables is given, and the theory is applied in Chapter 7 to the three-dimensional case of quadric surfaces. This latter chapter ends with a discussion of the symmetries of quadric surfaces. The first seven chapters of this text provide an exposition of the basic topics of solid analytic geometry, the material being adequate for a one-quarter course on the subject. The remain ing chapters contain additional material for longer courses or out side reading. Chapter 8, on spherical coordinates, contains a dis vi PREFACE cussion of some practical aspects of the theory of rotations and translations of axes in space. It is quite clear that rectangular coordinates are not as practical for actual measurements as are the coordinates of range, azimuth, and elevation, and in this chapter methods are developed for actual computation of the changes in these measurable coordinates after translations or rotations of axes. The chapter ends with a discussion of gno monic charts. Our final chapter contains a brief presentation of the elements of protective geometry. The effect on the theory of linear trans formations of the use of homogeneous coordinates is given, and the chapter contains a rigorous matrix proof of the invariance of the cross ratio under projective transformations. It is hoped that the use of modern algebraic techniques in this and in the earlier chapters of the present text will serve to make the subject of solid analytic geometry fit better in the teaching of modern mathematics than it has in the past. ADRIAN ALBERT CHICAGO, ILL. March, 1949 CONTENTS PREFACE v 1 COORDINATES AND LINES I 1. Vectors 1 2. Scalar multiplication 2 3. Inner products 3 4. The angle between two vectors 4 5. Directed lines 5 6. Orthogonal projections 6 7. Rectangular coordinates in ordinary space 7 8. The length of a vector in space 10 9. Lines through the origin 10 10. The angle between two vectors in space 12 11. Translation of axes 13 12... ... Read more

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ANALYTIO GEOMETRY INTRODUCTION DEFINITIONS AND FORMULAS FROM ALGEBRA AND TRIGONOMETRY 1. The Quadratic. - An expression reducible to the form AX2 + Bx + 0, where A, B, and ° are any constants, is called a quadratic mx. 2. Solution of the Quadratic. - The roots of any quadratic equation may be found by completing the square, or by using the quadratic formula. (a) To solve by completing the square, transpose the constant term to the right-hand member and divide both sides of the equation by the coefficient of x2? In the resulting equation add to both members the square of half the coefficient of x and extract the square root. (b) If the above rule is applied to the general quadratic AX2 + Bx + 0= 0, we get x= - B ±'JB2_4AO, 2A which is known as the quadmtic formula. To solve an equation by means of this formula simply substitute for A, B, and C in the formula their values in the given equation. 3. The Discriminant. - The expression under the radical, W - 4 AO, is

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Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry. ... Read more

Customer Reviews (1)

Great text for high school or college
I had used this text in a geometry class I took in college as a math major (class was not required, just needed hours to fill). After using this book and finishing the course, I found it to be much better than the geometry book I used in high school.

This book, like many high school texts, breaks geometry down into constructions using only a straight-edge and compass and then introduces definitions and theorems (some proved, some left to prove as exercises). The part that hooked me in was the fill-in-the-blank-step proofs that were partially completed but let the reader finish so that he or she will become familiar with "Statement/Reason" geometric proofs.

Finally, after ten chapters which deal with such topics as quadrilaterals, parallel lines, polygon areas, similar and congruent triangles, and circles, the authors introduce logic by methods of truth tables, Euler circles, and tautologies.

Last but not least, a chapter connecting algebra to 2-Dimensional (Cartesian) geometry offer the bridge between the "abstract" world of algebra and the "concrete" world of geometry. ... Read more

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This text has been adopted at:

University of Pennsylvania, PhiladelphiaUniversity of Connecticut, StorrsDuke University, Durham, NCCalifornia Institute of Technology, PasadenaUniversity of Washington, SeattleSwarthmore College, Swarthmore, PAUniversity of Chicago, ILUniversity of Michigan, Ann Arbor

"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."

- Revue Romaine de Mathematiques Pures et Appliquees

"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."

- Monatshefte F. Mathematik

"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."

- Publicationes Mathematicae

Contents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * Index

Series: Mathematics: Theory and Applications ... Read more

Customer Reviews (8)

Excelent book.
I studied the portuguese version of this book during the master degree program in mathematics at University of Brasilia, 1999. The book is very well written with beautiful results. Manfredo is an excellent mathematician, a great professor, and I had the chance to be present in many colloquiuns where he was the speaker. This is an very good exposition for those interested to learn more about the subject.

Needs a table of symbols
This is another well-written text by Do Carmo. I browsed through it and found I could not understand several passages because I did not know what the special symbols meant and there was no table of symbols. I plead with the publisher to add such a table to the next edition or printing.

Concise and clear
This is really a very good book to start Riemannian Geometry (RG). Exposition of key concepts of RG (affine connection, riemannian connection,geodesics, parallelism and sectional curvature, ...) are well motivated and concisely explained with numerous motivating and not so difficult execises. The book is self contained convenient for self study. It contains an introductory chapter on mathematical background explaining basic concepts as differentiable manifolds, immersion, embedding and so on, which are necessary to deal with RG. I have essentially one basic remark about this book. Formulation of RG as presented in it, is a little bit dated. Now, with the development of geometric algebra and Geometric calculus most, if not all, mathematical concepts needed to study RG like covariant derivative, curvature, and general tensors can be formulated without ressort to coordinates and in a manner to highlight their essential geometric features. Moreover derivation of certain formulae can be much easier and natural. For example the author defines the formula for |x^y| as sqrt(sqr(|x|).sqr(|y|)-sqr(inner product(x,y))). Then explains that it is the area of two dimensional parallelogram determined by the pair of vectors x and y. The reader might be puzzled as to how this formula is obtained. In the context of geometric algebra this is derived very naturally from basic concepts. Anyway, this remark does not diminish the value of this book.

Best 1st semester Riemannian Geometry book after 1 semester DG
This is the best Riemannian Geometry book after students have finished a semester of differential geometry.It gives geometric intuition, has plenty of exercises and
is excellent preparation for more advanced books like Cheeger-Ebin.

Students should already know differential geometry (Spivak "Calculus on manifolds" and Spivak "Differential Geometry Volume I" might be used there)

Warning: the curvature tensor is defined backwards as compared to Cheeger-Ebin.

Definitely a good start
This book is definitely a solid way to start in Riemannian geometry. The topics chosen give a glimpse of more advanced topics that the reader can venture to next, and the order covered leaves little confusion. The book is to the point, with little conversation about the concepts except at the very beginning of each chapter.

I only have two complaints, but neither would cause me to lower the rating to 4 stars.

1. There could be more "deep" exercises that allow the reader to explore more of the subtleties of the subject. And for what exercises there are, the author sometimes gives far too much away in "hints."

2. The book does not take a unified approach to the subject that fits nicely with the full generality of the theory. This is probably what makes the book good to start with, but there is still going to be a somewhat difficult transition from this book to a typical differential/riemannian geometry book. Namely, the basic language of vector bundles, pull backs/push forwards, tensors and tensor fields are either covered in a very specific framework or not at all. ... Read more

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The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem--the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most relevant contributions to the subject in the years 1959 to 1974. These include the pinching (or sphere) theorem, Berger's theorem for symmetric spaces, the differentiable sphere theorem, the structure of complete manifolds of non-negative curvature, and finally, results about the structure of complete manifolds of non-positive curvature. Emphasis is given to the phenomenon of rigidity, namely, the fact that although the conclusions which hold under the assumption of some strict inequality on curvature can fail when the strict inequality on curvature can fail when the strict inequality is relaxed to a weak one, the failure can happen only in a restricted way, which can usually be classified up to isometry. Much of the material, particularly the last four chapters, was essentially state-of-the-art when the book first appeared in 1975. Since then, the subject has exploded, but the material covered in the book still represents an essential prerequisite for anyone who wants to work in the field. ... Read more

Customer Reviews (2)

The basic reference
I am of course partial to the book ,I was graduate student at SUNY Stony Brook.
Still,I think this is the basic reference in the subject , a book that I like
to go back to review the most important theorems.
Clear, consise ,well written.
It is quite impissible to work in Riemanian Geometry without "Cheeger Ebin"

all you ever wanted to know about differential geometry
But thought the book was out of print. This classic book was published by North Holland in the late seventies, was photocopied by every grad student in geometry, and finally has been reprinted by the American Math Society. The book's goal (achieved) is to get you up to speed and working as quickly as possible. Riemannian geometry is covered from scratch (a la Milnor's Morse Theory (Annals of Mathematic Studies AM-51) but they don't stop there, and prove all of the basic comparison results. ... Read more

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The authors emphasize five objectives throughout the text: concreteness, readability, motivation, applicability and accuracy. The text includes over 600 problems, about 20 optional computer notes for supplementary reading by students motivated by computer and graphics calculator application, and a new section that illustrates applications of integration to the solution of separable differential equations. There are alsocomputer-generated figures to facilitate the visualization of surfaces and graphs of functions of two variables. ... Read more

Customer Reviews (1)

another good book on calculus
This is the typical first-year college textbook on calculus. Nothing especially remarkable about it. It may be a bit stronger on multivariate calculus than most first-year books, but that's about it. Colour figures are good. Explanations are clear. Proofs readable. Lot's of exercises with and without answers. And it includes some relatively silly BASIC programs to illustrate numerical methods. ... Read more

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College Calculus
One of the standard texts of the 60s & 70s that combined Calculus with Analytic Geometry.
Pre-requisites should be taken to fully understrand & appreciate the methodology. ... Read more

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Provides counter examples to the Osserman conjuncture in generic semi-Riemannian signature provided. The properties of semi-Riemannian Osserman manifolds are investigated. Softcover. ... Read more

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Excellent for engineering and technology students, this text goes far beyond instruction in standard orthographic projection to clarify all the tools of descriptive geometry--and how they apply to individual fields. The text places special emphasis on applications in all the various engineering disciplines: mechanical, plastics, industrial, piping, aerospace, marine, civil, and structural. As a result, students quickly grasp the value of descriptive geometry as they apply the tools and techniques to practical problems. By organizing information around the field's central concept--line of sight--the presentation facilitates understanding in a way unmatched by any other text. The worktext format provides students with all the resources they need--text and workbook--under one cover.ALSO AVAILABLEINSTRUCTOR SUPPLEMENTS CALL CUSTOMER SUPPORT TO ORDER Instructor's Guide, ISBN: 0-7668-0118-7Keywords: Descriptive Geometry ... Read more

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Applied Descriptive Geometry
I am unable to give an opinion as I had to return this item due to dropping the class it was needed for. ... Read more

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Riemann-Finsler geometry is a subject that concerns manifolds with Finsler metrics, including Riemannian metrics. It has applications in many fields of the natural sciences. Curvature is the central concept in Riemann–Finsler geometry. This invaluable textbook presents detailed discussions on important curvatures such as the Cartan torsion, the S-curvature, the Landsberg curvature and the Riemann curvature. It also deals with Finsler metrics with special curvature or geodesic properties, such as projectively flat Finsler metrics, Berwald metrics, Finsler metrics of scalar flag curvature or isotropic S-curvature, etc. Instructive examples are given in abundance, for further description of some important geometric concepts. The text includes the most recent results, although many of the problems discussed are classical. ... Read more

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A set in complex Euclidean space is called C-convex if all its intersections with complex lines are contractible, and it is said to be linearly convex if its complement is a union of complex hyperplanes. These notions are intermediates between ordinary geometric convexity and pseudoconvexity. Their importance was first manifested in the pioneering work of André Martineau from about forty years ago. Since then a large number of new related results have been obtained by many different mathematicians. The present book puts the modern theory of complex linear convexity on a solid footing, and gives a thorough and up-to-date survey of its current status. Applications include the Fantappié transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations. ... Read more