Introduction to Analytic Geometry

Introduction to Analytic Geometry tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩn...

AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY Philippe G. Ciarlet City University of Hong Kong Contents Preface 5 1 Three-dimensional differential geometry 9 Introduction 9 1.1 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Metrictensor 13 1.3 Volumes, areas, and lengths in curvilinear coordinates . . . . . . 16 1.4 Covariantderivativesofavectorfield 19 1.5 Necessary conditions satisfied by the metric tensor; the Riemann curvaturetensor 24 1.6 ExistenceofanimmersiondefinedonanopensetinR 3 with a prescribedmetrictensor 25 1.7 Uniqueness up to isometries of immersions with the same metric tensor 36 1.8 Continuity of an immersion as a function of its metric tensor . . 44 2 Differential geometry of surfaces 59 Introduction 59 2.1 Curvilinear coordinates on a surface . . . . . . . . . . . . . . . . 61 2.2 First fundamental form . . . . . . . . . . . . . . . . . . . . . . . 65 2.3 Areas and lengths on a surface . . . . . . . . . . . . . . . . . . . 67 2.4 Second fundamental form; curvature on a surface . . . . . . . . . 69 2.5 Principal curvatures; Gaussian curvature . . . . . . . . . . . . . . 73 2.6 Covariant derivatives of a vector field defined on a surface; the Gauß and Weingarten formulas . . . . . . . . . . . . . . . . . . . 79 2.7 Necessary conditions satisfied by the first and second fundamen- tal forms: the Gauß and Codazzi-Mainardi equations; Gauß’ TheoremaEgregium 82 2.8 Existence of a surface with prescribed first and second fundamen- talforms 85 2.9 Uniqueness up to proper isometries of surfaces with the same fundamental forms . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.10 Continuity of a surface as a function of its fundamental forms . . 100 3 4 Contents 3 Applications to three-dimensional elasticity in curvilinear coordinates 109 Introduction 109 3.1 The equations of nonlinear elasticity in Cartesian coordinates . . 112 3.2 Principle of virtual work in curvilinear coordinates . . . . . . . . 119 3.3 Equations of equilibrium in curvilinear coordinates; covariant derivativesofatensorfield 127 3.4 Constitutive equation in curvilinear coordinates . . . . . . . . . . 129 3.5 The equations of nonlinear elasticity in curvilinear coordinates . 130 3.6 The equations of linearized elasticity in curvilinear coordinates . 132 3.7 A fundamental lemma of J.L. Lions . . . . . . . . . . . . . . . . . 135 3.8 Korn’s inequalities in curvilinear coordinates . . . . . . . . . . . 137 3.9 Existence and uniqueness theorems in linearized elasticity in curvi- linearcoordinates 144 4 Applications to shell theory 153 Introduction 153 4.1 The nonlinear Koiter shell equations . . . . . . . . . . . . . . . . 155 4.2 The linear Koiter shell equations . . . . . . . . . . . . . . . . . . 164 4.3 Korn’sinequalitiesonasurface 172 4.4 Existence and uniqueness theorems for the linear Koiter shell equations; covariant derivatives of a tensor field defined on a surface 185 4.5 A brief review of linear shell theories . . . . . . . . . . . . . . . . 193 References 201 Index 209 PREFACE This book is based on lectures delivered over the years by the author at the Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at City University of Hong Kong. Its two-fold aim is to give thorough introduc- tions to the basic theorems of differential geometry and to elasticity theory in curvilinear coordinates. The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of anal- ysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations. In particular, no aprioriknowledge of differential geometry or of Introduction to Analytic Geometry Introduction to Analytic Geometry By: OpenStaxCollege 1/3 Introduction to Analytic Geometry (a) Greek philosopher Aristotle (384–322 BCE) (b) German mathematician and astronomer Johannes Kepler (1571–1630) The Greek mathematician Menaechmus (c 380–c 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular He presumed that the planets moved 2/3 Introduction to Analytic Geometry in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature His published law of planetary motion in the 1600s changed our view of the solar system forever He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane We will begin by studying each of three figures created in this manner We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems 3/3 Introduction to Differential Geometry & General Relativity TT TT hh hh ii ii rr rr dd dd PP PP rr rr ii ii nn nn tt tt ii ii nn nn gg gg JJ JJ aa aa nn nn uu uu aa aa rr rr yy yy 22 22 00 00 00 00 22 22 LL LL ee ee cc cc tt tt uu uu rr rr ee ee NN NN oo oo tt tt ee ee ss ss bb bb yy yy SS SS tt tt ee ee ff ff aa aa nn nn WW WW aa aa nn nn ee ee rr rr ww ww ii ii tt tt hh hh aa aa SS SS pp pp ee ee cc cc ii ii aa aa ll ll GG GG uu uu ee ee ss ss tt tt LL LL ee ee cc cc tt tt uu uu rr rr ee ee bb bb yy yy GG GG rr rr ee ee gg gg oo oo rr rr yy yy CC CC LL LL ee ee vv vv ii ii nn nn ee ee Departments of Mathematics and Physics, Hofstra University 2 Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds: An Introduction to Relativity 8. Covariant Differentiation 9. Geodesics and Local Inertial Frames 10. The Riemann Curvature Tensor 11. A Little More Relativity: Comoving Frames and Proper Time 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 13. Two Basic Premises of General Relativity 14. The Einstein Field Equations and Derivation of Newton's Law 15. The Schwarzschild Metric and Event Horizons 16. White Dwarfs, Neutron Stars and Black Holes, by Gregory C. Levine 3 1. Preliminaries Distance and Open Sets Here, we do just enough topology so as to be able to talk about smooth manifolds. We begin with n-dimensional Euclidean space E n = {(y 1 , y 2 , . . . , y n ) | y i é }. Thus, E 1 is just the real line, E 2 is the Euclidean plane, and E 3 is 3-dimensional Euclidean space. The magnitude, or norm, ||yy yy || of yy yy = (y 1 , y 2 , . . . , y n ) in E n is defined to be ||yy yy || = y 1 2 +y 2 2 +...+y n 2 , which we think of as its distance from the origin. Thus, the distance between two points yy yy = (y 1 , y 2 , . . . , y n ) and zz zz = (z 1 , z 2 , . . . , z n ) in E n is defined as the norm of zz zz - yy yy : Distance Formula Distance between and zz zz = ||zz zz yy yy || = (z 1 -y 1 ) 2 +(z 2 -y 2 ) 2 +...+(z n -y n ) 2 . Proposition 1.1 (Properties of the norm) The norm satisfies the following: (a) ||yy yy || ≥ 0, and ||yy yy || = 0 iff yy yy = 0 (positive definite) (b) ||¬yy yy || = |¬|||yy yy || for every ¬ é and yy yy é E n . (c) ||yy yy + zz zz || ≤ ||yy yy || + ||zz zz || for every yy yy , zz zz é E n (triangle inequality 1) (d) ||yy yy - zz zz || ≤ ||yy yy ww ww || + ||ww ww zz zz || for every yy yy ,, ,, zz zz ,, ,, ww ww é E n (triangle inequality 2) The proof of Proposition 1.1 is an exercise which may require reference to a linear algebra text (see “inner products”). Definition 1.2 A Subset U of E n is called open if, for every yy yy in U, all points of E n within some positive distance r of yy yy are also in U. (The size of r may depend on the point yy yy chosen. Illustration in class). Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set. Examples 1.3 (a) If a é E n , then the open ball with center aa aa and radius rr rr is the subset B(aa aa , r) = {x é E n | ||xx xx -aa aa || < r}. 4 Open balls are open sets: If xx xx é B(aa aa , r), ALGEBRAIC CURVES An Introduction to Algebraic Geometry WILLIAM FULTON January 28, 2008 Preface Third Preface, 2008 This text has been out of print for several years, with the author holding copy- rights. Since I continue to hear from young algebraic geometers who used this as their first text, I am glad now to make this edition available without charge to anyone interested. I am most grateful to Kwankyu Lee for making a careful LaTeX version, which was the basis of this edition; thanks also to Eugene Eisenstein for help with the graphics. As in 1989, I have managed to resist making sweeping changes. I thank all who have sent corrections to earlier versions, especially Grzegorz Bobi´nski for the most recent and thorough list. It is inevitable that this conversion has introduced some new errors, and I and future readers will be grateful if you will send any errors you find to me at wfulton@umich.edu. Second Preface, 1989 When this book first appeared, there were few texts available to a novice in mod- ern algebraic geometry. Since then many introductory treatises have appeared, in- cluding excellent texts by Shafarevich, Mumford, Hartshorne, Griffiths-Harris, Kunz, Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris. The past two decades have also seen a good deal of growth in our understanding of the topics covered in this text: linear series on curves, intersection theory, and the Riemann-Roch problem. It has been tempting to rewrite the book to reflect this progress, but it does not seem possible to do so without abandoning its elementary character and destroying its original purpose: to introduce students with a little al- gebra background to a few of the ideas of algebraic geometry and to help them gain some appreciation both for algebraic geometry and for origins and applications of many of the notions of commutative algebra. If working through the book and its exercises helps prepare a reader for any of the texts mentioned above, that will be an added benefit. i ii PREFACE First Preface, 1969 Although algebraic geometry is a highly developed and thriving field of mathe- matics, it is notoriously difficult for the beginner to make his way into the subject. There are several texts on an undergraduate level that give an excellent treatment of the classical theory of plane curves, but these do not prepare the student adequately for modern algebraic geometry. On the other hand, most books with a modern ap- proach demand considerable background in algebra and topology, often the equiv- alent of a year or more of graduate study. The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry, but without excessive prerequisites. We have assumed that the reader is familiar with some basic properties of rings, ideals, and polynomials, such as is often covered in a one-semester course in mod- ern algebra; additional commutative algebra is developed in later sections. Chapter 1 begins with a summary of the facts we need from algebra. The rest of the chapter is concerned with basic properties of affine algebraic sets; we have given Zariski’s proof of the important Nullstellensatz. The coordinate ring, function field, and local rings of an affine variety are studied in Chapter 2. As in any modern treatment of algebraic geometry, they play a funda- mental role in our preparation. The general study of affine and projective varieties is continued in Chapters 4 and 6, but only as far as necessary for our study of curves. Chapter 3 considers affine plane curves. The classical definition of the multiplic- ity of a point on a curve is shown to depend only on the local ring of the curve at the point. The [...]... proof of the theorem on factorization of integers into prime factors Irreducible components play the role of prime factors In view of Proposition 2, we can apply the previous terminology to a ne algebraic sets V Thus, we can speak about irreducible a ne algebraic k-sets, irreducible components of V and a decomposition of V into its irreducible components Notice that our topology depends very much... by means of the notion of a rational function on an a ne algebraic set First let us explain the condition that O(V ) is an integral domain We recall that V K n is a topological space with respect to the induced Zariski k-topology of K n Its closed subsets are a ne algebraic k-subsets of V From now on we denote by V (I ) the a ne algebraic k-subset of K n de ned by the ideal I k T ] If I = (F ) is... happens to be a k-algebra In particular, the eld R(V ) will be viewed as an extension k O(V ) R(V ) We will denote the eld of fractions of the polynomial ring k T1 ; : : : ; Tn ] by k(T1 ; : : : ; Tn ) It is called the eld of rational functions in n variables De nition A dominant rational k-map from an irreducible a ne algebraic k-set V to an irreducible a ne algebraic k-set W is a homomorphism of k-algebras... ) The corresponding factor-algebra k T ]=I (X ) is denoted by k X ] and is called the projective coordinate algebra of X The notion of a projective algebraic k-set is de ned similarly to the notion of an a ne algebraic k-set We x an algebraically closed extension K of k and consider subsets V Pn (K ) of the form PSol(S ; K ), where X is a system of homogeneous equations in n-variables with coe cients... on the set X (k) It assigns to a prime number p the image of m in Z=(p) = F p , i.e., the residue of m modulo p Now, we specialize the notion of a morphism of a ne algebraic varieties to de ne the notion of a regular map of a ne algebraic sets Recall that a ne algebraic k-set is a subset V of K n of the form X (K ), where X is an a ne algebraic variety over k and K is an algebraically closed extension... Prove that the correspondence K ! O(n; K ) ( = n n-matrices with entries in K satisfying M T = M ?1) is an abstract a ne algebraic k-variety 7 Give an example of a continuous map in the Zariski topology which is not a regular map 14 Irreducible algebraic sets 15 Lecture 4 IRREDUCIBLE ALGEBRAIC SETS AND RATIONAL FUNCTIONS We know that two a ne algebraic k-sets V and V 0 are isomorphic if and only if their... morphism of a ne algebraic k-varieties we have the following commutative diagram: fK ?! X (K ) = Homk (O(X ); K ) Y (K ) = Homk (O(Y ); K ) " ? ?" fO X X (O(X )) = Homk (O(X ); O(X )) ?! Y (O(X )) = Homk (O(Y ); O(X )): Take the identity map idO(X ) in the left bottom set It goes to the element in the left top set The bottom horizontal arrow sends idO(X ) to The right vertical arrow sends it to Now, because... )) = V (F ) An algebraic subsets of this form, where (F ) 6= f0g; (1), is called a hypersurface De nition A topological space V is said to be reducible if it is a union of two proper non-empty closed subsets (equivalently, there are two open disjoint proper subsets of V ) Otherwise V is said to be irreducible By de nition the empty set is irreducible An a ne algebraic k-set V is said to be reducible... must coincide with the image of under the top arrow, which is fK ( ) This proves the surjectivity The injectivity is obvious As soon as we know what is a morphism of a ne algebraic k-varieties we know how to de ne an isomorphism This will be an invertible morphism We leave to the reader to de ne the composition of morphisms and the identity morphism to be able to say Introduction to Differential Geometry & General Relativity 4th Printing January 2005 Lecture Notes by Stefan Waner with a Special Guest Lecture by Gregory C. Levine Departments of Mathematics and Physics, Hofstra University 2 Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries 3 2. Smooth Manifolds and Scalar Fields 7 3. Tangent Vectors and the Tangent Space 14 4. Contravariant and Covariant Vector Fields 24 5. Tensor Fields 35 6. Riemannian Manifolds 40 7. Locally Minkowskian Manifolds: An Introduction to Relativity 50 8. Covariant Differentiation 61 9. Geodesics and Local Inertial Frames 69 10. The Riemann Curvature Tensor 82 11. A Little More Relativity: Comoving Frames and Proper Time 94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 100 13. Two Basic Premises of General Relativity 109 14. The Einstein Field Equations and Derivation of Newton's Law 114 15. The Schwarzschild Metric and Event Horizons 124 16. White Dwarfs, Neutron Stars and Black Holes, by Gregory C. Levine131 References and Further Reading 138 3 1. Preliminaries Distance and Open Sets Here, we do just enough topology so as to be able to talk about smooth manifolds. We begin with n-dimensional Euclidean space E n = {(y 1 , y 2 , . . . , y n ) | y i é R}. Thus, E 1 is just the real line, E 2 is the Euclidean plane, and E 3 is 3-dimensional Euclidean space. The magnitude, or norm, ||y|| of y = (y 1 , y 2 , . . . , y n ) in E n is defined to be ||y|| = y 1 2 !+!y 2 2 !+!.!.!.!+!y n 2 , which we think of as its distance from the origin. Thus, the distance between two points y = (y 1 , y 2 , . . . , y n ) and z = (z 1 , z 2 , . . . , z n ) in E n is defined as the norm of z - y: Distance Formula Distance between y and z = ||z - y|| = (z 1 !-!y 1 ) 2 !+!(z 2 !-!y 2 ) 2 !+!.!.!.!+!(z n !-!y n ) 2 . Proposition 1.1 (Properties of the norm) The norm satisfies the following: (a) ||y|| ≥ 0, and ||y|| = 0 iff y = 0 (positive definite) (b) ||¬y|| = |¬|||y|| for every ¬ é R and y é E n . (c) ||y + z|| ≤ ||y|| + ||z|| for every y, z é E n (triangle inequality 1) (d) ||y - z|| ≤ ||y - w|| + ||w - z|| for every y, z, w é E n (triangle inequality 2) The proof of Proposition 1.1 is an exercise which may require reference to a linear algebra text (see “inner products”). Definition 1.2 A Subset U of E n is called open if, for every y in U, all points of E n within some positive distance r of y are also in U. (The size of r may depend on the point y chosen. Illustration in class). Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set. Examples 1.3 (a) If a é E n , then the open ball with center a and radius r is the subset B(a, r) = {x é E n | ||x-a|| < r}. 4 Open balls are open sets: If x é B(a, r), then, with s = r - ||x-a||, one has B(x, s) ¯ B(a, r). (b) E n is open. (c) Ø is open. (d) Unions of open sets are open. (e) Open sets are unions of open balls. (Proof in class) Definition 1.4 Now let M ¯ E s . A subset V ¯ M is called open in M (or relatively open) if, for every y in V, all points of M within some positive distance r of y are also in V. Examples 1.5 (a) Open balls in M If M ¯ E s , m é M, and r > 0, define B M (m, r) = {x é M | ||x-m|| < r}. Then B M (m, r) = B(m, r) Ú M, and so B M (m, r) is open in M. (b) M is open in M. (c) Ø is open in M. (d) Unions of open sets in M are open in ... that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular He presumed that the planets moved 2/3 Introduction to Analytic Geometry. . .Introduction to Analytic Geometry (a) Greek philosopher Aristotle (384–322 BCE) (b) German mathematician and astronomer Johannes... in this manner We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems 3/3