curve

A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The tangent bundle of a smooth manifold M (or indeed, any vector bundle over a manifold) is, at a fixed point, just a vector space and each such space can carry an inner product. If such a collection of inner products on the tangent bundle of a manifold vary smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve α(t): [0, 1] → M has tangent vector α′(t< ...

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Riemannian manifold, Riemannian manifold - Introduction

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Consider the curve defined by y = f(x) in a Cartesian coordinate system, and consider a point P with coordinates (c, f(c)) and another point Q with coordinates (c + Δx, f(c + Δx)). Then the slope m of the secant line, through P and Q, is given by: The righthand side of the above equation is a variation of Newton's difference quotient. As Δx approaches zero, this expression approaches the derivative ...

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Secant line, Secant line - How the secant function is related to secant lines, Secant line - Secant approximation

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Let there be a cone whose axis is the z-axis. Let its vertex be the origin. The equation for the cone is where and θ is the angle which the generators of the cone make with respect to the axis. Notice that this cone is actually a pair of cones: one cone standing upside down on the vertex of the other cone—or, as mathematicians say, this cone consists of two "nappes." Let there be a plane with a slope running along the x direction but which is level along the y direction. Its equation is < ...

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Conic section, Conic section - Types of conics, Conic section - Conics as point loci, Conic section - Eccentricity, Conic section - Conics in analytic geometry, Conic section - Semi-latus rectum and polar coordinates, Conic section - Properties, Conic section - Applications, Conic section - Dandelin spheres, Conic section - Derivation, Conic section - Derivation of the parabola, Conic section - Derivation of the ellipse, Conic section - Derivation of the hyperbola

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A one-dimensional polynomial spline, S(t), is an example of a piecewise function. In its most general form a polynomial spline, defined on an interval [a,b], consists of polynomial pieces, Pi(t), with each piece defined on one of a number of given subintervals . That is, It is required that the polynomial pieces on the subintervals all have degree n; and it is also required that tw ...

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Spline mathematics, Spline mathematics - Introduction, Spline mathematics - Definition, Spline mathematics - Examples, Spline mathematics - Notes, Spline mathematics - Representations and names, Spline mathematics - History

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The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametr ...

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Ellipse, Ellipse - Parametrisation, Ellipse - Eccentricity, Ellipse - Semi-latus rectum and polar coordinates, Ellipse - Area, Ellipse - Circumference, Ellipse - Stretching and Projection, Ellipse - Reflection property, Ellipse - Ellipses in physics, Ellipse - Ellipses in computer graphics

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An example of the second definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux. To better understand the concept of flux in Electromagnetism, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would ...

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Flux, Flux - Flux definition and theorems, Flux - Thermal systems, Flux - Chemical diffusion, Flux - Flux definition and theorems, Flux - Maxwell's equations, Flux - Poynting vector

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Let denote the Cantor space . We start with a continuous function h from the Cantor space onto the entire unit interval [0,1]. (The restriction of the Cantor function to the Cantor set is an example of such a function.) From it, we get a continuous function H from the topological product onto the entire unit square by setting See also:

Space-filling curve, Space-filling curve - Outline of the construction of a space-filling curve, Space-filling curve - The Hahn-Mazurkiewicz theorem, Space-filling curve - Literature

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As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M=H\Γ be a Riemann surface of negative ...

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Anosov diffeomorphism, Anosov diffeomorphism - Anosov flow on tangent bundles of Riemann surfaces, Anosov diffeomorphism - Lie vector fields, Anosov diffeomorphism - Anosov flow, Anosov diffeomorphism - Geometric interpretation of the Anosov flow, Anosov diffeomorphism - Historical references, Anosov diffeomorphism - Modern references

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In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the point with the same direction; such a line is the best straight-line approximation to the curve at that point. The curve, at point P, has the same slope as a tangent passing through P. The slope of a tangent line can be approximated by a secant line. It is a mistake to think of tangents as lines which intersect a curve at only one single point. There are tangents which intersect curves at several points ...

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Tangent, Tangent - Geometry, Tangent - Quote, Tangent - Related meaning, Tangent - Calculus, Tangent - Trigonometry, Tangent - Derivative

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Consider a function such that and (its derivative with respect to x) are continuous on [a,b]. The length s of the arc bounded by a and b is found by the formula . if function is defined parametrically where and . if function is defined via polar coordinates where r< ...

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Length of an arc, Length of an arc - Modern methods, Length of an arc - Historical methods, Length of an arc - Ancient, Length of an arc - 1600s, Length of an arc - Integral form

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Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M0 is a Cantor set. The Menger sponge is a closed set; since it is also bounded, the Heine-Borel theorem yields that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesgue measure 0. The topological dimension of the Menger sponge is one; indeed, the sponge was first constructed by Menger in 1926 while exploring the concept of topological dimensi ...

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Menger sponge, Menger sponge - Construction, Menger sponge - Properties, Menger sponge - Formal definition

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Formally, the winding number is defined as follows: If γ is a closed rectifiable curve in C, and z0 is a point in C not on γ, then the winding number of γ with respect to z0 (alternately called the index of γ with respect to z0) is defined by the formula: This is verifiable from applying the Cauchy integral formula — the integral will be a multiple of 2πi, since each time γ goes about z0, we have effectively calculated the integral again. ...

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Winding number, Winding number - Formal definitions, Winding number - Generalizations

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The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid. For example, the surface area of the torus with minor radius r and major radius R is ...

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Pappus's centroid theorem, Pappus's centroid theorem - The first theorem, Pappus's centroid theorem - The second theorem

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Having never thrown a pitch in the minor leagues, Pierce made his majors debut with the Detroit Tigers in 1945, just a few weeks after his 18th birthday. He was traded to the White Sox before the 1949 season. In 13 seasons with the Sox uniform, Pierce threw four one-hitters; pitched 51 consecutive scoreless innings and led the league in shutouts (7) and in strikeouts (186) in 1953; led the league in ERA (1.97) and in wins (20, along with Jim Bunning) in 1955; tied for the league lead in complete games between 1956 and 1958, and ...

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Billy Pierce, Billy Pierce - Career, Billy Pierce - Highlights, Billy Pierce - Inside the numbers

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An orbit can fail to be closed in two interesting ways. It could be an asymptotically periodic orbit if it converges to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit. An orbit can also be chaotic. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit. They exhibit sensitive dependence on initial conditions, meaning that small differences in the initial value will cause ...

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Orbit dynamics, Orbit dynamics - Closed Orbits

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Consider the unit circle S1 in R2. Then the map p : R → S1 with p(t) = (cos(t),sin(t)) is a cover where each point of S1 is covered infinitely often. Consider the complex plane with the origin removed, denoted by C×, and pick a non-zero integer n. Then p : C× → C× given by p(z) = zn is a cover ...

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Covering map, Covering map - Examples, Covering map - Elementary properties, Covering map - Universal covers, Covering map - Deck transformation group regular covers, Covering map - Monodromy action, Covering map - Group structure redux

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The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T, and the orthogonal group O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: where A is an orthogonal matrix or an orthogonal transformation followed by a translation: . T is a normal subgroup of E(n): for any translation t ...

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Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes

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Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x∈X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depen ...

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Covering map, Covering map - Examples, Covering map - Elementary properties, Covering map - Universal covers, Covering map - Deck transformation group regular covers, Covering map - Monodromy action, Covering map - Group structure redux

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If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x). More commonly, a function is defined by a formula, or more generally an algorithm — that is, a recipe that tells how to compute the value of f(x) given any x in the domain. See the squaring function sqr above. More generally, a function can also be defined by any mathematical condition relating the argument to the corresponding val ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Five of the most common forms of flux from the transport literature are defined as: Momentum flux, the rate of change of momentum moving across a unit area (N/m2). (Newtonian fluid, viscous flow) Heat flux, the rate of heat flow across a unit area (J/(m2 s)). (Fourier's Law) Chemical flux, the rate of movement of moles across a unit area (moles/(m2 s) ...

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Flux, Flux - Flux definition and theorems, Flux - Thermal systems, Flux - Chemical diffusion, Flux - Flux definition and theorems, Flux - Maxwell's equations, Flux - Poynting vector

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The graph of a function f is the set of all ordered pairs (x, f(x)), for all x in the domain X. If X and Y are the set of real numbers (or subsets thereof), then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of the plot's points There are theorems formulated or proved most eas ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Graph of a function