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Cover Postnikov M.M. Lectures in Geometry: Linear Algebra and Differential Geometry
Id: 7914
13.9 EUR

Lectures in Geometry: Linear Algebra and Differential Geometry. Semester 2

URSS. 320 pp. (English). Paperback. ISBN 978-5-88417-029-7.



This book is a direct continuation of the author's previous book* and is akin to it in being a nearly faithful record of the Lectures delivered by the author in the second semester of the first, year at the Mathematics-Mechanics Faculty of Moscow State University named after M. V. Lomonosov to mathematical students (a course in Linear Algebra and Analytic Geometry). Naturally, in the selection of the material and the order of presentation the author was guided by the same considerations as in the first semester (see the Preface in [1]). The number of Lectures in the book is explained by the fact that although the curriculum assigns 32 Lectures to the course, in practice it is impossible to deliver more than 27 Lectures.

The course in Linear Algebra and Analytic Geometry is just a part of a single two-year course in geometry, and much in this book is accounted for, as regards the choice of the material and its accentuation, by orientation to the second year devoted to the differential geometry of manifolds. In particular, it has proved possible (although it is not envisaged by the curriculum) to transfer part of the propaedeutic material of the third semester (the elementary differential geometry of curves and surfaces in three-dimensional space) to the second-semester course and this has substantially facilitated (not only for the Lecturer but, what is of course more important, also for the students) the third semester course. At the same time, as experience has shown, this material appeals to the students and they learn it well on the whole already in the second semester. M. M. Postnikov October 27, 1977


 Lecture 1 Vector spaces. Subspaces. Intersection of subspaces. Linear spans. A sum of subspaces. The dimension of a subspace. The dimension of a sum of subspaces. The dimension of a linear span
 Lecture 2 Matrix rank theorem. The rank of a matrix product. The Kronecker-Capelli theorem. Solution of systems of linear equations
 Lecture 3 Direct sums of subspaces. Decomposition of a space as a direct sum of subspacea. Factor spaces. Homomorphisms of vector spaces. Direct sums of spaces
 Lecture 4 The conjugate space. Dual spaces. A second conjugate space. The transformation of a conjugate basis and of the coordinates of covectors. Annulets. The space of solutions of a system of homogeneous linear equations
 Lecture 5 An annulet of an annulet and annulets of direct summands. Bilinear functionals and bilinear forms. Bilinear func-tionals in a conjugate space. Mixed bilinear functionals. Tensors
 Lecture 6 Multiplication of tensors. The basis of a space of tensors. Contraction of tensors. The rank space of a multilinear functional
 Lecture 7 The rank of a multilinear functional. Functionals and permutations. Alternation
 Lecture 8 Skew-symmetric multilinear functionals. External multiplication. Grassman algebra. External sums of covectors. Expansion of skew-symmetric functionals with respect to the external products of covectors of a basis
 Lecture 9 The basis of a space of skew-symmetric functionals. Formulas for the transformation of the basis of that space. Multivec-tors...The external rank of a skew-symmetric functional. Multi vector rank theorem. Conditions for the equality of multivectors
 Lecture 10 Cartan/s divisibility theorem. Pliicker relations. The Plu-cker coordinates of subspaces. Planes in an affine space. Planes in a projective space and their coordinates
 Lecture 11 Symmetric and skew-symmetric bilinear functionals. A matrix of symmetric bilinear functionals. The rank of a bilinear functional. Quadratic functionals and quadratic forms. Lagrange theorem
 Lecture 12 Jacobi theorem. Quadratic forms over the fields of complex and real numbers. The law of inertia. Positively definite quadratic functionals and forms
 Lecture 13 Second degree hypersurfaces of an n-dimensional projective space. Second degree hypersurfaces in a complex and a real-complex projective space. Second degree hypersurfaces of an n-dimensional affine space. Second degree nypersurfaces in a complex and a real-complex affine space
 Lecture 14 The algebra of linear operators. Operators and mixed bilinear functionals. Linear operators and matrices. Invertible operators. The adjoint operator. The Fredholm alternative. Invariant subspaces and induced operators
 Lecture 15 Eigenvalues. Characteristic roots. Diagonalizable operators. Operators with simple spectrum. The existence of a basis in which the matrix of an operator is triangular. Nilpo-tent operators
 Lecture 16 Decomposition of a nilpotent operator as a direct sum of cyclic operators. Root subspaces. Normal Jordan form. The Hamilton-Cayley theorem
 Lecture 17 Complexificationof a linear opera tor. Proper subspaces belonging to characteristic roots. Operators whose complexifica-tion is diagonalizable
 Lecture 18 Euclidean and unitary spaces. Orthogonal complements. The identification of vectors and covectors. Annulets and orthogonal complements. Bilinear functionals and linear operators. Elimination of arbitrariness in the identification of tensors of different types. The metric tensor. Lowering and raising of indices
 Lecture 19 Adjoint operators. Self-adjoint operators. Skew-symmetric and skew-Hermitian operators. Analogy between Hermitian operators and real numbers. Spectral properties of self-adjoint operators. The orthogonal diagonalizability of self-adjoint operators
 Lecture 20 Bringing quadratic forms into canonical form by orthogonal transformation of variables. Second degree hypersurfaces in a Euclidean point space. The minimax property of eigenvalues of self-adjoint operators. Orthogonally diagonalizable operators
 Lecture 21 Positive operators. Isometric operators. Unitary matrices. Polar factorization of invertible operators. A geometrical interpretation of polar factorization. Parallel translations and centroaffine transformations. Bringing a unitary operator into diagonal form. A rotation of an ^-dimensional Euclidean space as a composition of rotations in two-dimensional planes
 Lecture 22 Smooth functions. Smooth hypersurfaces. Gradient. Derivatives with respect to a vector. Vector fields. Singular points of a vector field. A module of vector fields. Potential and ir-rotational vector fields. The rotation of a vector field. The divergence of a vector field. Vector analysis. Hamilton's symbolic vector. Formulas for products. Compositions of operators
 Lecture 23 Continuous, smooth, and regular curves. Equivalent curves. Regular curves in the plane and graphs of functions. The tangential hyperplane of a 'hypersurface. The length of a curve. Curves in the plane. Curves in a three-dimensional space
 Lecture 24 Projections of a curve onto the coordinate planes of the moving n-hedron. Prenet's formulas for a curve in the n-dimen-sional space. Representation of a curve by its curvatures. Regular surfaces. Examples of surfaces
 Lecture 25 Vectors tangential to a surface. The tangential plane. The first quadratic form of a surface. Mensuration of lengths and angles on a surface. Diffeomorphismsof surfaces. Isometries and the intrinsic geometry of a surface. Examples. Developables
 Lecture 26 The tangential plane and the normal vector. The curvature of a normal section. The second quadratic form of a surface. The indicatfix of Dupin. Principal curvatures. The second quadratic form of a graph. Ruled surfaces of zero curvature. Surfaces of revolution
 Lecture 27 Weingarten's derivation formulas. Coefficients of connection. The Gauss theorem. The necessary and sufficient conditions of isometry
 Subject Index

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