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从高观点看初等数学/Elementary mathematics from an advanced standpoint

从高观点看初等数学/Elementary mathematics from an advanced standpoint

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作者: Felix Klein 著

出版社: Oversea Publishing House

出版时间: 2004-06

版次: 1

ISBN: 9780486434810

定价: 117.07

装帧: 平装

开本: 其他

纸张: 胶版纸

页数: 214页

分类: 外文古旧书>英文书>社会文化

2人买过

This text begins with the simplest geometric manifolds, the Grassmann determinant principle for the plane and the Grassmann principle for space; and more. Also explores affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. Concludes with a systematic discussion of geometry and its foundations. 1939 edition. 141 figures. Part One: The Simplest Geometric Manifolds

  I. Line-Segment, Area, Volume, as Relative Magnitudes

    Definition by means of determinants; interpretation of the sign

    Simplest applications, especially the cross ratio

    Area of rectilinear polygons

　　Curvilinear areas

　　Theory of Amsler's polar planimeter

　　Volume of polyhedrons, the law of edges

　　One-sided polyhedrons

　II. The Grassmann Determinant Principle for the Plane

    Line-segment (vectors)

    Application in statics of rigid systems

    Classification of geometric magnitudes according to their behavior under trans formation of rectangular coordinates

    Application of the principle of classification to elementary magnitudes

　III. The Grassmann Principle for Space

　　Line-segment and plane-segment

　　Application to statics of rigid bodies

　　Relation to MSbius' null-system

　　Geometric interpretation of the null-system.

　　Connection with the theory of screws

　IV. Classification of the Elementary Configurations of Space according to their

    Behavior under Transformation of Rectangular Coordinates

    Generalities concerning transformations of rectangular space coordinates

    Transformation formulas for some elementary magnitudes

    Couple and free plane magnitude as equivalent manifolds

    Free line-segment and free plane magnitude ("polar" and "axial" vector)

　　Scalars of first and second kind

　　Outlines of a rational vector algebra

　　Lack of a uniform nomenclature in vector calculus

　V. Derivative Manifolds

　  Derivatives from points (carves, surfaces, point sets)

    Difference between analytic and synthetic geometry

    Projective geometry and the principle of duality

    Pliicker's analytic method and the extension of the principle of duality (lin coordinates)

    Grassmann's Ausdehnungsiehre;n-dimensional geometry

    Scalar and vector fields; rational vector analysis

Part Two: Geometric Transformations

　Transformations and their analytic representation

　I. AtBne Transformations

　  Analytic definition and fundamental properties

    Application to theory of ellipsoid

    Parallel projection from one plane upon another

    Axonometric mapping of space (affine transformation with vanishing deter- minant)

    Fundamental theorem of Poblke

　II. Projective Transformations

    Analytic definition; introduction of homogeneous coordinates

    Geometric definition: Every coUineation is a projective transformation

    Behavior of fundamental manifolds under projective transformation

    Central projection of space upon a plane (projective transformation with　vanishing determinant)

    Relief perspective

    Application of projection in deriving properties of comcs

　III. Higher Point Transformations

  　1. The Transformation by Reciprocal Radii

　   Peaucellier's method of drawing a line

　   Stereographic projection of the sphere

　　2. Some More General Map Projections.

　   Mercator's projection

　   Tissot theorems

　　3. The Most General Reversibly Unique Continuous Point Transformatins

　   Genus and connectivity of surfaces

　   Euler's theorem on polyhedra

　IV. Transformations wlth Change of Space Element

    1. Dualistic Transformations

    2. Contact Transformations

    3. Some Examples

     Forms of algebraic order and class curves

     Application of contact transformations to theory of cog wheels

　V. Theory of the Imaginary

　  Imaginary cirde-points and imaginary sphere-circle

    Imaginary transformation

    Von Staudt s interpretation of self-conjugate imaginary manifolds by means oJ real polar systems

    Von Staudt's complete interpretation of single imaginary elements

    Space relations of imaginary points and lines

……

Part Three:Systematic Discussion of Geometry and Its Foundations

  II Foundations of Geometry

  Index of Names

  Index of Contents
内容简介:
This text begins with the simplest geometric manifolds, the Grassmann determinant principle for the plane and the Grassmann principle for space; and more. Also explores affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. Concludes with a systematic discussion of geometry and its foundations. 1939 edition. 141 figures.
目录:
Part One: The Simplest Geometric Manifolds

  I. Line-Segment, Area, Volume, as Relative Magnitudes

    Definition by means of determinants; interpretation of the sign

    Simplest applications, especially the cross ratio

    Area of rectilinear polygons

　　Curvilinear areas

　　Theory of Amsler's polar planimeter

　　Volume of polyhedrons, the law of edges

　　One-sided polyhedrons

　II. The Grassmann Determinant Principle for the Plane

    Line-segment (vectors)

    Application in statics of rigid systems

    Classification of geometric magnitudes according to their behavior under trans formation of rectangular coordinates

    Application of the principle of classification to elementary magnitudes

　III. The Grassmann Principle for Space

　　Line-segment and plane-segment

　　Application to statics of rigid bodies

　　Relation to MSbius' null-system

　　Geometric interpretation of the null-system.

　　Connection with the theory of screws

　IV. Classification of the Elementary Configurations of Space according to their

    Behavior under Transformation of Rectangular Coordinates

    Generalities concerning transformations of rectangular space coordinates

    Transformation formulas for some elementary magnitudes

    Couple and free plane magnitude as equivalent manifolds

    Free line-segment and free plane magnitude ("polar" and "axial" vector)

　　Scalars of first and second kind

　　Outlines of a rational vector algebra

　　Lack of a uniform nomenclature in vector calculus

　V. Derivative Manifolds

　  Derivatives from points (carves, surfaces, point sets)

    Difference between analytic and synthetic geometry

    Projective geometry and the principle of duality

    Pliicker's analytic method and the extension of the principle of duality (lin coordinates)

    Grassmann's Ausdehnungsiehre;n-dimensional geometry

    Scalar and vector fields; rational vector analysis

Part Two: Geometric Transformations

　Transformations and their analytic representation

　I. AtBne Transformations

　  Analytic definition and fundamental properties

    Application to theory of ellipsoid

    Parallel projection from one plane upon another

    Axonometric mapping of space (affine transformation with vanishing deter- minant)

    Fundamental theorem of Poblke

　II. Projective Transformations

    Analytic definition; introduction of homogeneous coordinates

    Geometric definition: Every coUineation is a projective transformation

    Behavior of fundamental manifolds under projective transformation

    Central projection of space upon a plane (projective transformation with　vanishing determinant)

    Relief perspective

    Application of projection in deriving properties of comcs

　III. Higher Point Transformations

  　1. The Transformation by Reciprocal Radii

　   Peaucellier's method of drawing a line

　   Stereographic projection of the sphere

　　2. Some More General Map Projections.

　   Mercator's projection

　   Tissot theorems

　　3. The Most General Reversibly Unique Continuous Point Transformatins

　   Genus and connectivity of surfaces

　   Euler's theorem on polyhedra

　IV. Transformations wlth Change of Space Element

    1. Dualistic Transformations

    2. Contact Transformations

    3. Some Examples

     Forms of algebraic order and class curves

     Application of contact transformations to theory of cog wheels

　V. Theory of the Imaginary

　  Imaginary cirde-points and imaginary sphere-circle

    Imaginary transformation

    Von Staudt s interpretation of self-conjugate imaginary manifolds by means oJ real polar systems

    Von Staudt's complete interpretation of single imaginary elements

    Space relations of imaginary points and lines

……

Part Three:Systematic Discussion of Geometry and Its Foundations

  II Foundations of Geometry

  Index of Names

  Index of Contents

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