After the Russian edition of this book appeared some of my fellow Lecturers asserted that many of the Lectures in the book are far too long to be physically delivered during the allowed two teaching periods. By the right of friendship I had to remind them that a Lecturer must prepare for his Lectures—even if he has been lecturing for over a dozen years— and make in advance an elaborate, practically minute-by-minute plan of every Lecture. It is necessary to consider eforehand the rhythm of the Lecture to be delivered—what portions of it are to be read slowly, almost at dictation speed, and what may be said quicker—and its pattern of intonation—where to raise the voice and where to lower it. One also needs a joke somewhere about the middle of the Lecture to rouse the tired students and it should be prepared yet at home, and in every detail, up to a play of facial muscles. It goes without saying that one must plan in advance what to write on the blackboard and in what order and where, and when to delete anything, and coordinate all this with everything else. It is surprising how all this extends the limits of Lecture time and how much it is then possible to say in an outwardly unhurried and thorough manner, with numerous repetitions and explanations.
Some reviewers have reproached me for a systematic use of bivectors and trivectors saying that one may well do without them. Some well-known physicist, Max Planck, I think, once said that new ideas (he meant scientific ideas but this can be fully applied to methodical ideas as well), could win only when their opponents have retired from the stage as a result of a natural change of generations. An ex- cellent example illustrating this thesis is the introduction of vectors into the courses of analytic geometry half a century ago. Now only a few people remember the fierce discussions concerning this matter and the present generation does not know how many a lance was broken and how much ink split in attempts to prove that vectors were a harmful thing because replacing three equations in coordinates by one vector equation they saved paper but proportionally hampered comprehension. The last of the authoritative opponents of vectors in the USSR died soon after the war but some ten years more passed before diffidently excusatory reservations disappeared altogether from vectorial presentations of geometry (as well as from mechanics and physics where, however, this happened a little earlier). Now bivectors and tri-vectors are awaiting their turn.
I have taken the opportunity to introduce some minor improvements in the text. The most serious one is perhaps a simpler construction of the complexification of an affine space in Lecture 19. It is true that it contains a certain element of arbitrariness (which was what restrained me at first) but experience has shown that this arbitrariness is perfectly harmless. Besides, the last Lecture has been divided into two since two versions of the concluding Lecture were combined in it for purely technical, internal editorial reasons. So the book contains 29 Lectures now.
As far as I can judge with my poor knowledge of English the translation is well done and conveys all the nuances of my thought. M.M. Postnikov May 1, 1980 Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29
Preface to the English edition
The subject-matter of analytic geometry. Vectors. Vector addition. Multiplication of a vector by a number. Vector spaces. Examples. Vector spaces over an arbitrary field
The simplest consequences of the vector space axioms. Independence of the sum of any number of vectors on brackets arrangement. The concept of a family
Linear dependence and linear independence. Linearly independent sets. The simplest properties of linear dependence. Linear-dependence theorem.
Collinear vectors. Coplanar vectors. The geometrical meaning of collinearity and coplanarity. Complete families of vectors, bases, dimensionality. Dimensionality axiom. Basis criterion. Coordinates of a vector. Coordinates of the sum of vectors and those of the product of a vector by a number.
Isomorphisms of vector spaces. Coordinate isomorphisms. The. isomorphism of vector spaces of the same dimension. The method of coordinates. Affine spaces. The isomorphism of affine spaces of the same dimension. Affine coordinates. Straight lines in affine space. Segments.
Parametric equations of a straight line. The equation of a straight line in a plane. The canonical equation of a straight line in a plane. The general equation of a straight line in a plane. Parallel lines. Relative position of two straight lines in a plane. Uniqueness theorem. Position of a straight line relative to coordinate axes. The half-planes into which a straight line divides a plane.
An intuitive notion of a bivector. A formal definition of the bivector. The coincidence of the two definitions. A zero bivector. Conditions for the equality of bivectors. Parallelism of the vector and the bivector. The role of the three-dimensionality condition. Addition of bivectors.
The correctness of the definition of a bivector sum. The product of a bivector by a number. Algebraic properties of external product. The vector space of bivectors. Bivectors in a plane and the theory of areas. Bivectors in space.
Planes in space. Parametric equations of a plane. The general equation of a plane. A plane passing through three noncollinear points.
The half-spaces into which a plane divides space. Relative positions of two planes in space. Straight lines in space. A plane containing a given straight line and passing through a given point. Relative positions of a straight line and a plane in space. Relative positions of two straight lines in space. Change from one basis for a vector space to another.
Formulas for the transformation of vector coordinates. Formulas for the transformation of the affine coordinates of points. Orientation. Induced orientation of a straight line. Orientation of a straight line given by an equation. Orientation of a plane in space.
Deformation of bases. Sameness of the sign bases. Equivalent bases and matrices. The coincidence of deformabil-ity with the sameness of sign. Equivalence of linearly independent systems of vectors. Trivectors. The product of a trivector by a number. The external product of three vectors.
Trivectors in three-dimensional vector space. Addition of trivectors. The formula for the volume of a parallelepiped. Scalar product. Axioms of scalar multiplication. Euclidean spaces. The length of a vector and the angle between vectors. The Cauchy-Buniakowski inequality. The triangle inequality. Theorem on the diagonals of a parallelogram. Orthogonal vectors and the Pythagorean theorem.
Metric form and metric coefficients. The condition of positive definiteness. Formulas for the transformation of metric coefficients when changing a basis. Orthonormal families of vectors and Fourier coefficients. Orthonormal bases and rectangular coordinates. Decomposition of positive definite matrices. The Gram-Schmidt orthogonalization process. Isomorphism of Euclidean spaces. Orthogonal matrices. Second-order orthogonal matrices. Formulas for he transformation of rectangular coordinates.
Trivectors in oriented Euclidean space. Triple product of three vectors. The area of a bivector in Euclidean space. A vector complementary to a bivector in oriented Euclidean space. Vector multiplication. Isomorphism of spaces of vectors and bivectors. Expressing a vector product in terms of coordinates. The normal equation of a straight line in the Euclidean plane and the distance between a point and a straight line. Angles between two straight lines in the Euclidean plane.
The plane in Euclidean space. The distance from a point to a plane. The angle between two planes, between a straight line and a plane, between two straight lines. The distance from a point to a straight line in space. The distance between two straight lines in space. The equations of the common perpendicular of two skew lines hn space.
The parabola. The ellipse. The focal and directorial properties of the ellipse. The hyperbola. The focal and directorial properties of the hyperbola.
The equations of ellipses, parabolas and hyperbolas leferred to a vertex. Polar coordinates. The equations of ellipses, parabolas and hyperbolas in polar coordinates. Affine ellipses, parabolas, hyperbolas. Algebraic curves. Second-degree curves and associated difficulties. Complex affine geometry and its insufficiency.
Real-complex vector spaces. Their dimensionality. Isomorphism of real-complex vector spaces. Complexification. Real-complex affine spaces. The complexification of affine spaces. Real-complex Euclidean spaces. Real and imaginary curves of second degree.
Introductory remarks. The centre of a second-degree curve. Centres of symmetry. Central and nonceniral curves of second degree. Straight lines of non-asymptotic direction. Tangents. Straight lines of asymptotic direction.
Singular and nonsingular directions. Diameters. Diameters and centres. Conjugate directions and conjugate diameters. Simplification of the equation of the second-degree central curve. Necessary refinements. Simplification of the equation of the second-degree rioncentral curve.
Second-degree curves in the complex affine plane. Second-degree curves in the real-complex affine plane. The uniqueness of the equation of a second-degree curve. Second-degree curves in the Euclidean plane. Circles.
Ellipsoids. Imaginary ellipsoids. Second-degree imaginary cones. Hyperboloids of two sheets. Hyperboloids of one sheet. Rectilinear generators of a hyperboloid of one sheet. Second-degree cones. Elliptical paraboloids. Hyporbolic paraboloids. Elliptical cylinders. Other second-degree surfaces. The statement of the classification theorem.
Coordinates of a straight line. Pencils of straight lines. Ordinary and ideal pencils. Extended planes. Models of projective-affine geometry.
Homogeneous affine coordinates. Equations of straight lines in homogeneous coordinates. Second-degree curves in the projective-affine plane. Circles in the projective-Euclidean real-complex plane. Projective planes. Homogeneous afftne coordinates in the bundle of straight lines. Formulas for the transformation of homogeneous affine coordinates. Projective coordinates. Second-degree curves in the projec-tive plane.
Coordinate isomorphisms of vector spaces. Coordinate isomorphisms of affine spaces. Projective-affine spaces. Projective spaces. Pencils of planes. Bundles of planes. Extending space with ideal elements. Orthogonal, affine and projective transformations.
Expressing an affine transformation in terms of coordinates. Examples of affine transformations. Factorization of affine transformations. Orthogonal transformations. Motions of a plane. Symmetries and glide symmetries. A motion of a plane as a composition of two symmetries. Rotations of a space.
The Desargues theorem. The Pappus-Pascal theorem. The Fano theorem. The duality principle. Models of the projective plane. Models of the projective straight line and of the projective space. The complex projective straight line.
Linear fractional transformations. Linear transformations. Inversion. Inversions and linear fractional transformations. Two properties of linear fractional transformations. Fixed points of linear fractional transformations. Parabolic, elliptical, hyperbolic and loxodromic linear fractional transformations. The three-point theorem. The multiplier of linear fractional nonparabolic transformation. Classification of linear fractional transformations. Stereographic projection formulas. Rotations of a sphere as linear fractional transformations of a plane. Isometries of a cube Subject index.
Una Asombrosa Aventura de Jules 01
Diccionario de Onomatopeyas del Cómic
Pepe 01
Oscuro
Algebraic Geometry
Lectures In Geometry: Lie Groups And Lie Algebras (semester V)