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Fundamental Theorem Of Curves
In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion.. Use A curve can be described, and thereby defined, by a pair of scalar fields: curvature \kappa and torsion \tau, both of which depend on some parameter which parametrizes the curve but which can ideally be the arc length of the curve. From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve. Congruence If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent to each other. See also *Differential geometry of curves *Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Arc Length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating speed, the magnitude of the velocity vector over the curve with respect to time. Thus the length of a continuously differentiable curve (x(t),y(t)), for a\le t\le b, in the Euclidean plane is given as the integral L = \int_a^b \sqrt\,dt, (because \sqrt is the magnitude of the velocity vector (x'(t),y'(t)), i.e., the particle's speed). The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length. Determining the length of an irregular arc segment by approximating the arc segment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured “within” or along the surface, not on the way it is isometrically embedding, embedded in Euclidean space. This is the content of the ''Theorema Egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema Egregium'' in 1827. Informal definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Differential Geometry Of Curves
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Frenet–Serret Formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \R^3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. The tangent, normal, and binormal unit vectors, often called , , and , or collectively the Frenet–Serret basis (or TNB basis), together form an orthonormal basis that spans \R^3, and are defined as follows: * is the unit vector tangent to the curve, pointing in the direction of mot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vector Field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parametric Equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a Surface (mathematics), surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (also spelled parametrization, parametrisation) of the object. For example, the equations \begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the unit circle, where is the parameter: A point is on the unit circle if and only if there is a value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Scalar Field
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. Definition Mathematically, a scalar field on a region ''U'' is a real or complex-valued function or distribution on ''U''. The region ''U'' may be a set in some Euclidean space, Minkowski spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
