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Affine Differential Geometry
Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under volume-preserving affine transformations. The name affine differential geometry follows from Klein's Erlangen program. The basic difference between affine and Riemannian differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics.Contents1 Preliminaries 2 The first Induced volume form 3 The second induced volume form 4 Two natural conditions 5 The conclusion 6 The affine normal line6.1 Plane curves 6.2 Surfaces in 3-space7 See also 8 ReferencesPreliminaries[edit] Here we consider the simplest case, i.e
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Differential Form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan
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Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain.Contents1 Differentiability classes1.1 Examples 1.2 Multivariate differentiability classes 1.3 The space of Ck functions2 Parametric continuity2.1 Definition 2.2 Order of continuity3 Geometric continuity3.1 Smoothness
Smoothness
of curves and surfaces 3.2 Smoothness
Smoothness
of piecewise defined curves and surfaces4 Other concepts4.1 Relation to analyticity 4.2 Smooth partitions of unity 4.3 Smooth functions between manifolds 4.4 Smooth functions between subsets of manifolds5 See also 6 ReferencesDifferentiability classes[edit] Differentiability class is a classification of functions according to the properties of their derivatives
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Surface Normal
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality. The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns: [ 1 9 − 13 20 5 − 6 ] . displaystyle begin bmatrix 1&9&-13\20&5&-6end bmatrix
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Scalar Product
In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates
Cartesian coordinates
of two vectors is widely used and often called inner product (or rarely projection product); see also inner product space. Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces
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Parallel Transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection. The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport
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Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian differential and integral geometer. His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner. In 1916 Blaschke published one of the first books devoted to convex sets: Circle and Sphere (Kreis und Kugel). Drawing on dozens of sources, Blaschke made a thorough review of the subject with citations within the text to attribute credit in a classical area of mathematics. In 1933 Blaschke signed the Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State. Wilhelm Blaschke
Wilhelm Blaschke
joined the NSDAP
NSDAP
in 1937.[1]Contents1 Publications 2 Namesake 3 References 4 External linksPublications[edit]Kreis und Kugel, Leipzig, Veit 1916; 3rd edn. Berlin, de Gruyter 1956 Vorlesungen über Differentialgeometrie, 3 vols., Springer, Grundlehren der mathematischen Wissenschaften 1921-1929 (vol
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Special Linear Group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det : GL ⁡ ( n , F ) → F × . displaystyle det colon operatorname GL (n,F)to F^ times
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Semi-direct Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an outer semidirect product is a Cartesian product as a set, but with a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (a.k.a
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Lie Group
In mathematics, a Lie group
Lie group
(pronounced /liː/ "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory
Galois theory
for analysing the discrete symmetries of algebraic equations
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Open Interval
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers R displaystyle mathbb R , the set of all negative real numbers, and the empty set. Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define
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Inflexion Point
In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. If the curve is the graph of a function y = f(x), of differentiability class C2, this means that the second derivative of f vanishes and changes of sign at the point
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
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Tangent Line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz
Leibniz
defined it as the line through a pair of infinitely close points on the curve.[1] More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f '(c) where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point
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Parallel Line
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet. Parallel lines are the subject of Euclid's parallel postulate.[1] Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry
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