In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: (x2 + y2)2 = a2(x2 − y2) Graphing this equation produces a curve similar to . The curve has become a symbol of infinity and is widely used in math. The symbol itself is sometimes referred to as the lemniscate. Its Unicode represe ...

Including:

• Lemniscate - Other equations
• Lemniscate - Arc length and elliptic functions

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In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short e ...

Including:

• 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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In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. In the fields of heat transport and mass transport (fluid dynamics, hydrogeology, chemical engineering), flux is defined as the amount of a given quantity that flows through a unit area per unit time (Bird, Stewart, and Lightfoot, Transport Phenomenon, 1960). Flux in this definition is a vector. In the field of electromagnetism, flux is usually the integral of a vector quantity o ...

Including:

• 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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In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. More literally, a cusp is a sharp point or apex, such as occurs in two dimensions at the end of a crescent, or in three dimensions at the tip of a cone or horn. There are a number of technical terms derived from it. In mathematics, a cusp might mean a singular point of a curve such as that seen near (0,0) for y = x2/3;

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Vector field - Gradient field. Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on X(a scalar field) such that The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero. Vector field - C ...

See also:

Vector field, Vector field - Definition, Vector field - Notes, Vector field - Examples, Vector field - Gradient field, Vector field - Central field, Vector field - Curve integral, Vector field - Flow curves, Vector field - Difference between scalar and vector field, Vector field - Example 3

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Contravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor. Another method is used to derive covariant tensor components. When performing tensor transformations it is critical that the method used to map to the coordinate systems in use be tracked s ...

See also:

Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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A lemniscate may also be described by the polar equation r2 = a2cos2φ or the bipolar equation ...

See also:

Lemniscate, Lemniscate - Other equations, Lemniscate - Arc length and elliptic functions

Read more here: » Lemniscate: Encyclopedia II - Lemniscate - Other equations

On a (pseudo-)Riemannian manifold M a geodesic is defined as a smooth curve γ(t) that parallel transports its own tangent vector. That is, . where ∇ stands for the Levi-Civita connection on M. In the case of a Riemannian manifold, the geodesics that one obtains this way are identical to geodesics for the induced metric space. In terms of local coordinates on M the geodesic equation can be written (using the summation convention): where xa(t) are the coordinates of t ...

See 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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The cissoid of Diocles is named after the Greek geometer Diocles who used it in 180 B.C. to solve the Delian problem: how much must the length of a cube be increased in order to double the volume of the cube? Diocles' solution is correct, except that the solution involves an intersection of a line with a construction of a cissoid of Diocles, which cannot be accomplished by means of the simple but strict Greek rules of compass and straightedge constructio ...

See also:

Cissoid of Diocles, Cissoid of Diocles - Construction, Cissoid of Diocles - Delian problem, Cissoid of Diocles - Roulette, Cissoid of Diocles - The cissoid of Diocles as a pedal curve, Cissoid of Diocles - Inversion

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One can ask what happens, when a stationary observer is in orbit just outside the event horizon and (against all advice) sticks his hand through the horizon? The answer is: he won't succeed in doing so. Free orbits are only possible at a certain distance (for a non-rotating black hole, this figure is at least three times the Schwarzschild radius). Near the event horizon, an observer can only remain at a constant radius when he uses a force (e.g. from a rocket) to keep him there. The force needed grows to infinity when the observer wants to m ...

See also:

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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Levi-Civita connection defines also a derivative along curves, usually denoted by D. Given a smooth curve γ on (M,g) and a vector field V on γ its derivative is defined by ...

See also:

Levi-Civita connection, Levi-Civita connection - Formal definition, Levi-Civita connection - Derivative along curve

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The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as developed later in the article. Lie derivative - The Lie derivative of a function. One might start by defining the Lie derivative in terms of the differential of a function. Thus, given a function and a vector field X defined on M, one defines the Lie derivative of f at point as the usual derivative ...

See also:

Lie derivative, Lie derivative - Definition, Lie derivative - The Lie derivative of a function, Lie derivative - The Lie derivative of a vector field, Lie derivative - The Lie derivative of differential forms, Lie derivative - Properties, Lie derivative - Lie derivative of tensor fields, Lie derivative - Coordinate expressions, Lie derivative - Generalizations, Lie derivative - Nijenhuis-Lie derivative

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Some authors use the extended real number line and allow the distance function d to attain the value ∞. Such a metric is called an extended metric. Every extended metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equiva ...

See also:

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 Koch Curve can be completely described as an Lindenmayer System using the following definition: Angle: π/3 (60°) Axiom: F Rules: F → F-F++F-F In addition, the Koch Snowflake can be defined as follows: Angle: π/3 (60°) Axiom: F++F++F Rules: F → F-F++F-F ...

See also:

Koch curve, Koch curve - L-System Definition, Koch curve - Implementation, Koch curve - External link

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Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves. The idea of an indifference curve is a straightforward one: If a consumer was equally satisfied with 1 apple and 4 bananas, 2 apples and 2 bananas, or 5 apples and 1 banana, these combinations would all lie on the same indifference curve. Indifference curve - Preference Relations. Suppose that the set of alternatives among which a consumer can choose is called . Den ...

See also:

Indifference curve, Indifference curve - History, Indifference curve - Preference Relations and Utility, Indifference curve - Preference Relations, Indifference curve - Formal link to Utility theory, Indifference curve - Indifference Curve Properties, Indifference curve - Indifference Map, Indifference curve - Assumptions, Indifference curve - Examples of Indifference Curves, Indifference curve - Application

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Graph of a function - Hardware. Graphing calculator Oscilloscope Graph of a function - Software. See List of graphing software ...

See also:

Graph of a function, Graph of a function - Tools for plotting function graphs, Graph of a function - Hardware, Graph of a function - Software

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The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

See also:

Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, 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 - 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

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Introduction

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant. Any straight line through the origin will intersect a logarithmic spiral at the same angle α, which can be computed (in radians) as arctan(1/ln(b)). The pitch angle of the spiral is the (constant) angle the spiral makes with circles centered at the origin. It can be computed as ar ...

See also:

Logarithmic spiral, Logarithmic spiral - Definition, Logarithmic spiral - Notes, Logarithmic spiral - Logarithmic spirals in nature

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The path integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, γ : [a, b] → U is a rectifiable curve and f : U → C is a function. Then the path integral may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression The integral is then the limit of this sum, as the l ...

See also:

Path integral, Path integral - Complex analysis, Path integral - Example, Path integral - Vector calculus, Path integral - Definition, Path integral - Path independence, Path integral - Applications, Path integral - Relationship with the path integral in complex analysis, Path integral - Quantum mechanics

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Consider the one-dimensional scalar conservation equation ut + fx(u) = 0. Here u and f are scalar, with u(x,t) a function of x and t. By the chain rule this equation implies that ut + fuSee also:

Method of characteristics, Method of characteristics - Example, Method of characteristics - Bibliography

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The first mathematical formulation of gravity was Isaac Newton's law of universal gravitation, published in his 1687 work Principia Mathematica. Professor William Whewell of Cambridge University, author of History of the Inductive Sciences (1837) stated: "The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of the truth disclosed, or the fundamental and satisfactory nature of this truth." [In A Treasury o ...

See also:

Gravity, Gravity - Overview of the history of gravitational theory, 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 - Newton's law of universal gravitation, Gravity - Acceleration due to gravity, Gravity - Bodies with spatial extent, Gravity - Vector form, Gravity - Gravitational field, 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

Read more here: » Gravity: Encyclopedia II - Gravity - Overview of the history of gravitational theory