ManifoldTopological space that locally resembles Euclidean spaceThe Klein bottle immersed in three-dimensional spaceThe surface of the Earth requires (at least) two charts to include every point without plotting the same point more than once on the same chart. Here the globe is decomposed into charts around the North and South Poles.In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
Surface (topology)In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensionalsolid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.Curve orientationIn mathematics, an orientation of a curve (including polygonal curves) is the choice of one of the two possible senses for travelling on the curve, as in forward and backward. For example, for Cartesian coordinates on the Euclidean plane, the x-axis is traditionally oriented toward the right, and the y-axis is upward oriented. At each point along the curve, there are two tangent directions possible, colinear and opposite to each other.
SphereA sphere (from Ancient Greek (sphaîra)'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and the distance r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics.
Euclidean spaceEuclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimensionn, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
Topological spaceIn mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Loop (topology)In mathematics, a loop in a topological spaceX is a continuous functionf from the unit intervalI = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. A loop may also be seen as a continuous map f from the pointedunit circleS into X, because S may be regarded as a quotient of I under the identification of 0 with 1.
Vector spaceAlgebraic structure in linear algebraVector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
ClockwiseTwo-dimensional rotation can occur in two possible senses or "directions" of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to the left, and back up to the top. The opposite sense of rotation or revolution is (in Commonwealth English) anticlockwise (ACW) or (in North American English) counterclockwise (CCW).
Jordan curve theoremIn topology, a discipline within mathematics, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every simple closed curve in the plane (known as a Jordan curve) divides the plane into two regions: the interior, bounded by the curve, and an unbounded exterior, containing all of the nearby and faraway exterior points. Every continuous path connecting a point of one region to a point of the other must intersect the curve somewhere.
CurveIn 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.*As an Amazon Associate I earn from qualifying purchases.