Encyclopedia - Ordinary Differential Equation: Encyclopedia - Ordinary Differential Equation
In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of ...   » Read the article

Encyclopedia - Length Of An Arc: Encyclopedia - Length Of An Arc
Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. Although many met...   » Read the article

Encyclopedia - Billy Pierce: Encyclopedia - Billy Pierce
Walter William (Billy) Pierce (born April 2, 1927 in Detroit, Michigan) is a former starting pitcher in Major League Baseball. From 1945...   » Read the article

Encyclopedia - Calculus: Encyclopedia - Calculus
Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus Product rule |...   » Read the article

Encyclopedia - Anosov Diffeomorphism: Encyclopedia - Anosov Diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain t...   » Read the article

Encyclopedia - Angle: Encyclopedia - Angle
An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat...   » Read the article

Encyclopedia - Gottfried Leibniz: Encyclopedia - Gottfried Leibniz
Gottfried Wilhelm von Leibniz (also Leibnitz) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German poly...   » Read the article

Encyclopedia - Spiral: Encyclopedia - Spiral
In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, de...   » Read the article

Encyclopedia - Émile Clapeyron: Encyclopedia - Émile Clapeyron
Benoit Paul Émile Clapeyron (February 26, 1799 - January 28, 1864) was an French engineer and physicist, one of the founders of thermody...   » Read the article

Encyclopedia - Line Mathematics: Encyclopedia - Line Mathematics
A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathem...   » Read the article

Encyclopedia - Vector Field: Encyclopedia - Vector Field
In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector ...   » Read the article

Encyclopedia - Continuous Function: Encyclopedia - Continuous Function
In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in ...   » Read the article

Encyclopedia - Contour Line: Encyclopedia - Contour Line
A contour line (also level set, isopleth, isogram or isarithm) for a function of two variables is a curve connecting points where the fun...   » Read the article

Encyclopedia - Gravity: Encyclopedia - Gravity
Gravity is a force of attraction that acts between bodies that have mass. It is a physical phenomenon of fundamental importance, profoun...   » Read the article

Encyclopedia - Conic Section: Encyclopedia - Conic Section
In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surfac...   » Read the article

Encyclopedia - Cone: Encyclopedia - Cone
A cone is a basic geometrical shape: see cone (solid). Several things have also been called "cones" on account of their shape: A volcani...   » Read the article

Encyclopedia - Chord Geometry: Encyclopedia - Chord Geometry
A chord of a curve is a geometric line segment whose endpoints both lie on the curve. A secant or a secant line is the line extension of ...   » Read the article

Encyclopedia - Mechanical Work: Encyclopedia - Mechanical Work
Work (abbreviated W) is the energy transferred by a force to a moving object. Work is a scalar quantity, but it can be positive or negati...   » Read the article

Encyclopedia - Chloracne: Encyclopedia - Chloracne
Chloracne is an acne-like eruption of blackheads, cysts, and pustules associated with over-exposure to certain halogenic aromatic hydroca...   » Read the article

Encyclopedia - Bullet Rogan: Encyclopedia - Bullet Rogan
Charles Wilber(n) "Bullet" Rogan, a.k.a. "Bullets" or "Bullet Joe" (July 28, 1893 - March 4, 1967), born in Oklahoma City, Oklahoma, play...   » Read the article

Encyclopedia - Lemniscate: Encyclopedia - Lemniscate
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...   » Read the article

Encyclopedia - Ellipse: Encyclopedia - Ellipse
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 c...   » Read the article

Encyclopedia - Flux: Encyclopedia - Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. In the ...   » Read the article

Encyclopedia - Cusp: Encyclopedia - Cusp
In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. More ...   » Read the article

Encyclopedia - Vector Field: Encyclopedia Ii - Vector Field - Examples
Vector field - Gradient field. Vector fields can be constructed out of scalar fields using the vector operator gradient which gives ris...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - What 'contravariant' Means
Contravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projec...   » Read the article

Encyclopedia - Lemniscate: Encyclopedia Ii - Lemniscate - Other Equations
A lemniscate may also be described by the polar equation r2 = a2cos2φ or the bipolar equation ... See also:Lemniscate, Lemniscate...   » Read the article

Encyclopedia - Geodesic: Encyclopedia Ii - Geodesic - Pseudo-riemannian Geometry
On a (pseudo-)Riemannian manifold M a geodesic is defined as a smooth curve γ(t) that parallel transports its own tangent vector. That i...   » Read the article

Encyclopedia - Cissoid Of Diocles: Encyclopedia Ii - Cissoid Of Diocles - Delian Problem
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 l...   » Read the article

Encyclopedia - Event Horizon: Encyclopedia Ii - Event Horizon - Sticking Your Hand Through An Event Horizon
One can ask what happens, when a stationary observer is in orbit just outside the event horizon and (against all advice) sticks his hand ...   » Read the article

Encyclopedia - Levi-civita Connection: Encyclopedia Ii - Levi-civita Connection - Derivative Along Curve
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 ...   » Read the article

Encyclopedia - Lie Derivative: Encyclopedia Ii - Lie Derivative - Definition
The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie deriva...   » Read the article

Encyclopedia - Metric Mathematics: Encyclopedia Ii - Metric Mathematics - Related Concepts And Alternative Axiom Systems
Some authors use the extended real number line and allow the distance function d to attain the value ∞. Such a metric is called an exte...   » Read the article

Encyclopedia - Koch Curve: Encyclopedia Ii - Koch Curve - L-system Definition
The Koch Curve can be completely described as an Lindenmayer System using the following definition: Angle: π/3 (60°) Axiom: F Rules: ...   » Read the article

Encyclopedia - Indifference Curve: Encyclopedia Ii - Indifference Curve - Preference Relations And Utility
Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves. The idea...   » Read the article

Encyclopedia - Graph Of A Function: Encyclopedia Ii - Graph Of A Function - Tools For Plotting Function Graphs
Graph of a function - Hardware. Graphing calculator Oscilloscope Graph of a function - Software. See List of graphing software .....   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Introduction
The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler...   » Read the article

Encyclopedia - Logarithmic Spiral: Encyclopedia Ii - Logarithmic Spiral - Notes
The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarith...   » Read the article

Encyclopedia - Path Integral: Encyclopedia Ii - Path Integral - Complex Analysis
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 cu...   » Read the article

Encyclopedia - Method Of Characteristics: Encyclopedia Ii - Method Of Characteristics - Example
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. B...   » Read the article

Encyclopedia - Gravity: Encyclopedia Ii - Gravity - Overview Of The History Of Gravitational Theory
The first mathematical formulation of gravity was Isaac Newton's law of universal gravitation, published in his 1687 work Principia Mathe...   » Read the article

Encyclopedia - Euclidean Group: Encyclopedia Ii - Euclidean Group - Subgroups
Types of subgroups of E(n): Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two wh...   » Read the article

Encyclopedia - Félix Hernández: Encyclopedia Ii - Félix Hernández - Discovery As A Prospect
Hernández was first spotted by Luis Fuenmayor, a part-time Mariners scout who saw him pitching at age 14 in a tournament near Maracaibo,...   » Read the article

Encyclopedia - Lebesgue Covering Dimension: Encyclopedia Ii - Lebesgue Covering Dimension - Some Unusual Topological Constructions
The definition of the Lebesgue covering dimension can be used to build some unusual topological sets, such as the Sierpinski carpet. A co...   » Read the article

Encyclopedia - Jacobian: Encyclopedia Ii - Jacobian - Jacobian Matrix
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact tha...   » Read the article

Encyclopedia - Ordinary Differential Equation: Encyclopedia Ii - Ordinary Differential Equation - Types Of Differential Equations With Some History
The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Ricca...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - The Vocabulary Of Functions
An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the ...   » Read the article

Encyclopedia - Gravity: Encyclopedia Ii - Gravity - Overview Of The History Of Gravitational Theory
The first mathematical formulation of gravity was Isaac Newton's law of universal gravitation, published in his 1687 work Principia Mathe...   » Read the article

Encyclopedia - Spiral: Encyclopedia Ii - Spiral - Two-dimensional Spirals
A two-dimensional spiral may be described using polar coordinates by saying that the radius r is a continuous monotonic function of θ. T...   » Read the article

Encyclopedia - Émile Clapeyron: Encyclopedia Ii - Émile Clapeyron - Work
Émile Clapeyron - Thermodynamics. In 1834, he made his first contribution to the creation of modern thermodynamics by publishing a rep...   » Read the article

Encyclopedia - Covering Map: Encyclopedia Ii - Covering Map - Elementary Properties
Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every there exists an open set A in C containing...   » Read the article

Encyclopedia - Angle: Encyclopedia Ii - Angle - Conventions On Measurement
A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, an...   » Read the article

Encyclopedia - Tile: Encyclopedia Ii - Tile - Roof Tiles
Roof tiles are designed mainly to keep out rain, and are traditionally made from locally available materials such as clay, slate, or wood...   » Read the article

Encyclopedia - Laffer Curve: Encyclopedia Ii - Laffer Curve - Critiques Of The Laffer Curve
Conventional economic paradigms acknowledge the basic notion of the Laffer curve, but argue that government was operating on the left-han...   » Read the article

Encyclopedia - Continuous Function: Encyclopedia Ii - Continuous Function - Real-valued Continuous Functions
Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the three functions h, T and M ...   » Read the article

Encyclopedia - Mechanical Work: Encyclopedia Ii - Mechanical Work - Types Of Work
Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electric...   » Read the article

Encyclopedia - Quaternions And Spatial Rotation: Encyclopedia Ii - Quaternions And Spatial Rotation - Introducion
The 1909 edition of Webster's unabridged dictionary (citation: Webster's New International Dictionary, G. & C. Merriam Co. copyright ...   » Read the article

Encyclopedia - Tangent Space: Encyclopedia Ii - Tangent Space - Formal Definitions
There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite st...   » Read the article

Encyclopedia - Calculus: Encyclopedia Ii - Calculus - Differential Calculus
The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula: for an object movi...   » Read the article

Encyclopedia - Tom Gordon: Encyclopedia Ii - Tom Gordon - Major League Baseball Career
Gordon started his career as a Royal, and was signed away by Boston where he was converted from a starter to a closer. In 1998, Gordon se...   » Read the article

Encyclopedia - Chloracne: Encyclopedia Ii - Chloracne - Etiology And Progression
Chloracne normally results from direct skin contact with chloracnegens, although ingestion and inhalation are also possible causative rou...   » Read the article

Encyclopedia - Roulette Curve: Encyclopedia Ii - Roulette Curve - Example
Fixed curve is a catenary, rolling curve is a line: f(t) = t + icosh(t)   f'(t) = 1 + isinh(t) r(t) = sinh(t)   r'(t) = cosh(t...   » Read the article

Encyclopedia - Line Mathematics: Encyclopedia Ii - Line Mathematics - Line Segment
In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end point...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Mathematical Definition Of A Function
A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an eleme...   » Read the article

Encyclopedia - Riemannian Manifold: Encyclopedia Ii - Riemannian Manifold - Introduction
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 ea...   » Read the article

Encyclopedia - Secant Line: Encyclopedia Ii - Secant Line - Secant Approximation
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 po...   » Read the article

Encyclopedia - Conic Section: Encyclopedia Ii - Conic Section - Derivation
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 ...   » Read the article

Encyclopedia - Spline Mathematics: Encyclopedia Ii - Spline Mathematics - Definition
A one-dimensional polynomial spline, S(t), is an example of a piecewise function. In its most general form a polynomial spline, defined o...   » Read the article

Encyclopedia - Ellipse: Encyclopedia Ii - Ellipse - Parametrisation
The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor ...   » Read the article

Encyclopedia - Flux: Encyclopedia Ii - Flux - Flux Definition And Theorems
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 cros...   » Read the article

Encyclopedia - Space-filling Curve: Encyclopedia Ii - Space-filling Curve - Outline Of The Construction Of A Space-filling Curve
Let denote the Cantor space . We start with a continuous function h from the Cantor space onto the entire unit interval [0,1]. (The res...   » Read the article

Encyclopedia - Anosov Diffeomorphism: Encyclopedia Ii - Anosov Diffeomorphism - Anosov Flow On Tangent Bundles Of Riemann Surfaces
As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This f...   » Read the article

Encyclopedia - Tangent: Encyclopedia Ii - Tangent - Geometry
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 dire...   » Read the article

Encyclopedia - Length Of An Arc: Encyclopedia Ii - Length Of An Arc - Modern Methods
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...   » Read the article

Encyclopedia - Menger Sponge: Encyclopedia Ii - Menger Sponge - Properties
Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of th...   » Read the article

Encyclopedia - Winding Number: Encyclopedia Ii - Winding Number - Formal Definitions
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 ...   » Read the article

Encyclopedia - Pappus's Centroid Theorem: Encyclopedia Ii - Pappus's Centroid Theorem - The First Theorem
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 ...   » Read the article

Encyclopedia - Billy Pierce: Encyclopedia Ii - Billy Pierce - Career
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 hi...   » Read the article

Encyclopedia - Orbit Dynamics: Encyclopedia Ii - Orbit Dynamics - Closed Orbits
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....   » Read the article

Encyclopedia - Covering Map: Encyclopedia Ii - Covering Map - Examples
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 cove...   » Read the article

Encyclopedia - Euclidean Group: Encyclopedia Ii - Euclidean Group - Subgroup Structure Matrix And Vector Representation
The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T, and the orthogon...   » Read the article

Encyclopedia - Covering Map: Encyclopedia Ii - Covering Map - Monodromy Action
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 bel...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Specifying A Function
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...   » Read the article

Encyclopedia - Flux: Encyclopedia Ii - Flux - Flux Definition And Theorems
There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with dis...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Graph Of A Function
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 (o...   » Read the article

Encyclopedia - Flux: Encyclopedia Ii - Flux - Chemical Diffusion
Flux, or diffusion, for gaseous molecules can be related to the function: where N is the total number of gaseous particles, k is Boltz...   » Read the article

Encyclopedia - Covering Map: Encyclopedia Ii - Covering Map - Deck Transformation Group Regular Covers
A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all...   » Read the article

Encyclopedia - Covering Map: Encyclopedia Ii - Covering Map - Universal Covers
A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important property: if p ...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Algebra And Geometry
In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a ...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Is A Function More Than Its Graph?
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 codo...   » Read the article

Encyclopedia - Cissoid Of Diocles: Encyclopedia Ii - Cissoid Of Diocles - Construction
Choose a point O and a line L not passing through O. Point O and line L define the plane in which the cissoid will be drawn. Draw a line ...   » Read the article

Encyclopedia - Cissoid Of Diocles: Encyclopedia Ii - Cissoid Of Diocles - Roulette
This curve is also a roulette. Take two congruent parabolas, set them vertex-to-vertex, and roll one along the other; the vertex of the r...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Category Theory
The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects ...   » Read the article

Encyclopedia - Euclidean Group: Encyclopedia Ii - Euclidean Group - Relation To The Affine Group
The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product struc...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Classes Of Functions
Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-o...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Partial Functions And Multi-functions
The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x i...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Pointwise Operations
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can defi...   » Read the article

Encyclopedia - Cissoid Of Diocles: Encyclopedia Ii - Cissoid Of Diocles - The Cissoid Of Diocles As A Pedal Curve
THEOREM: The pedal curve of a parabola with respect to its vertex is a cissoid of Diocles. Proof: Any parabola can be rotated and transla...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Informal Usage
In common physics usage, the adjective covariant may sometimes be used informally as a synonym for invariant (or equivariant, in mathemat...   » Read the article

Encyclopedia - Continuous Function: Encyclopedia Ii - Continuous Function - Continuous Functions Between Topological Spaces
Main article: continuity (topology) The above definitions of continuous functions can be generalized to functions from one topological sp...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Composing Functions
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then apply...   » Read the article