curve

So Newton's original formula was: where the symbol means "is proportional to". To make this into an equal-sided formula or equation, there needed to be a multiplying factor or constant that would give the correct force of gravity no matter the value of the masses or distance between them. This gravitational constant was discovered in 1797 by Henry Cavendish. Thus the discovery and application of Newton's law of gravity accounts for the detailed information we have about the planets in our sol ...

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Gravity, Gravity - Overview of the history of gravitational theory, Gravity - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, Gravity - The Earth's gravity, Gravity - Comparative gravities of the Earth Sun Moon and planets, Gravity - Mathematical equations for a falling body, Gravity - Gravitational potential, Gravity - Acceleration relative to the rotating Earth, Gravity - Gravity and astronomy, Gravity - Self-gravitating system, Gravity - Practical uses of gravity, Gravity - Problems with Newton's theory, Gravity - Theoretical concerns, Gravity - Disagreement with observation, Gravity - Newton's reservations, Gravity - Einstein's theory of gravitation, Gravity - Experimental tests, Gravity - Comparison with electromagnetic force, Gravity - Gravity and quantum mechanics, Gravity - Alternative theories, Gravity - Recent alternative theories, Gravity - Historical alternative theories, Gravity - Notes

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Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, some mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain. There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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 set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the fact that |YX| = |Y||X|. See the article on cardinal numbers for more details. It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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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If f is a function from X to Y, the set X is called the domain of f, and Y is called its codomain. Each element of the domain is called an argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x by (or under) the function. The value of a function f at an argument x is traditionally written f(xSee also:

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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Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one quantity is completely determined by another quantity. Thus, for example, one may use a function to describe how the temperature of water affects its density. Functions are also used in computer science to model data structures and the effects of algorithms. However, the word is also used in computing in the very different sense of pro ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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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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. More generally, a function can be defined by any mathematical condition relating the argument to the corresponding value. There are many other ways of defining functio ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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 - Specifying a function

Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets. For example, consider the multiplication function that associates two integers to their product: f(x, y) = x·y. This function can be defined formally as having domain Z ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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 - Functions with multiple inputs and outputs

Another use of a Euclidean group is for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in E+(3). The Euclidean groups are Lie groups, so that calculus notions can be adapted immediately from this setting. ...

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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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For a given set X two metrics d1 and d2 are called topological equivalent (uniformly equivalent) if the identity mapping id: (X,d1) → (X,d2) is a homeomorphism (uniform isomorphism). ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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These conditions express intuitive notions about the concept of distance. For example, that the distance between distinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequality means that the distance traversed directly between x and z, is not larger than the distance to traverse in going first from x to y, and then from y to z. Euclid in his work proved that the shortest distance between two points is a line; that w ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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Given a normed vector space (X,||.||) we can define a metric on X by d(x,y):=||x-y||. The metric d is called induced by ||.||. Conversely if a metric d on a vector space X satisfies the properties d(x,y) = d(x+a,y+a) (translation invariance) d(αx,αy) = |α|d(x,y) (homogenity) then we can define a ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

Read more here: » Metric mathematics: Encyclopedia II - Metric mathematics - Relation of norms and metrics

Event horizons also exist in the absence of gravity. A simple example is a uniform accelerated particle (whose speed will thus eventually approach the speed of light but will always be smaller). Light emitted at a certain distance in the direction of that particle will never reach the accelerated particle. It is beyond the event horizon for that particle. Such event horizons occur in particle accelerators. A part of spacetime forms an event horizon as observed from a constantly accelerated observer. The world line of the observ ...

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Event horizon, Event horizon - Sticking your hand through an event horizon, Event horizon - Event horizon in the absence of gravity, Event horizon - Other examples of an event horizon, Event horizon - External link

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The terms function, mapping, map and transformation are usually used synonymously. It is not necessary that we be able to display the explicit formula for a function. For example, your postal code is a function of the location of your residence. If a function is given by a formula, then the formula is stated when the function is first introduced, and must be referred back to every time that function is used. For example ...

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Function mathematics, Function mathematics - Symbols and language, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - Functions of more than one variable, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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A metric on a set X is a function (called the distance function or simply distance) d : X × X → R (where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions: d(x, y) ≥ 0     (non-negativity) d(x, y) = 0   if and only if   x = y     (identity ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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The shortest path between two points in a curved space can be found by writing the equation for the length of a curve, and then minimizing this length using standard techniques of calculus and differential equations. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the r ...

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Geodesic, Geodesic - Introduction, Geodesic - Examples, Geodesic - Metric geometry, Geodesic - pseudo-Riemannian geometry, Geodesic - Existence and uniqueness, Geodesic - Geodesic flow, Geodesic - Geodesic spray, Geodesic - Variations and generalizations

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In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from the unit interval I to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t1, t2See also:

Geodesic, Geodesic - Introduction, Geodesic - Examples, Geodesic - Metric geometry, Geodesic - pseudo-Riemannian geometry, Geodesic - Existence and uniqueness, Geodesic - Geodesic flow, Geodesic - Geodesic spray, Geodesic - Variations and generalizations

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In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form then: if h2 = ab, the equation represents a parabola; if h2 < ab and a b and/or h0 , the equation represents an ellipse; if h2 > ab, the equation represents a hyperbola; if h2See also:

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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Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing t ...

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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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The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to a and b by the formula , or . In polar coordinates, a conic section with one focus at the origin and, if any, the other on the positive x-axis, is given by the equation . ...

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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

Read more here: » Conic section: Encyclopedia II - Conic section - Semi-latus rectum and polar coordinates

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For specific applications of each type of conic section, see the arti ...

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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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Note: Readers not familiar with multivariate calculus or vectors, please see "Simpler formulae" below Work is defined as the following line integral: where: C is the path or curve traversed by the object; is the force vector; is the position vector. This formula readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero (viz. circu ...

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Mechanical work, Mechanical work - Definition, Mechanical work - Units, Mechanical work - Simpler formulae, Mechanical work - Types of work, Mechanical work - PV work, Mechanical work - Mechanical energy, Mechanical work - Conservation of mechanical energy

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In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric optics a ray or a (light) beam is a line or curve that describes the direction in which light or other electromagnetic radiation is propagated. The ray is perpen ...

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Line mathematics, Line mathematics - Line segment, Line mathematics - Ray

Read more here: » Line mathematics: Encyclopedia II - Line mathematics - Ray