从高观点看初等数学/Elementary mathematics from an advanced standpoint 网盘下载 pdf 免费 2025 在线 epub 电子版

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

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