manifolds

There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; ...

See also:

Connectedness, Connectedness - Other notions of connectedness, Connectedness - Connectivity

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In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'qua ...

Including:

• Cohomology - History
• Cohomology - Cohomology theories
• Cohomology - Eilenberg-Steenrod theories
• Cohomology - Extraordinary cohomology theories
• Cohomology - Other cohomology theories

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Stratification is the building up of layers of deposits, and can have several variations of meaning: Social stratification, is the dividing of a society into levels based on wealth or power. Stratification in archaeology are the layers in which objects are found. Stratification (botany). See stratified sampling for the use of stratification in survey sampling. The term "stratified sampling" is also refers a method of variance reduction in Monte Carlo methods. In logic, stratifica

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In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd(S), fr(S), and . There are several common (and equivalent) definitions to the boundary of ...

Including:

• Boundary topology - Examples
• Boundary topology - Properties
• Boundary topology - Boundary of a boundary

Read more here: » Boundary topology: Encyclopedia - Boundary topology

Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates Integration by substitution | Integration by parts | Integration by trigonometric substitution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals In calculus, the chain rule is a formula for th ...

Including:

• Chain rule - The general power rule
• Chain rule - Example I
• Chain rule - Example II
• Chain rule - Chain rule for several variables
• Chain rule - Proof of the chain rule
• Chain rule - The fundamental chain rule
• Chain rule - Tensors and the chain rule

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Dr. Brian Greene (born February 9, 1963) is a physicist and one of the world's foremost string theorists. Since 2003 he has been a professor at Columbia University. Born in New York City, Greene was a prodigy in mathematics. At the age of five, he could multiply 30-digit numbers. His skill in mathematics was such that by the time he was twelve years old, he was being privately tutored in mathematics by a Columbia University professor because he had surpassed the high-school math level. He entered Harvard in 1980 to major in physics, and with his bachelor's degree, Greene went to ...

Including:

• Brian Greene - Facts
• Brian Greene - Important contributions to physics
• Brian Greene - Publications

Read more here: » Brian Greene: Encyclopedia - Brian Greene

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of the manifold's tangent bundle. V ...

Including:

• 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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The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

Including:

• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

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Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. Algebraic topology - The method of algebraic invariants. The goal is to take topological spaces and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for ...

Including:

• Algebraic topology - The method of algebraic invariants
• Algebraic topology - Results on homology
• Algebraic topology - Setting in category theory
• Algebraic topology - Applications of algebraic topology

Read more here: » Algebraic topology: Encyclopedia - Algebraic topology

Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. Although much of his working life was spent in Zürich and then Princeton, he is closely identified with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as pure disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and a key member of the Institute for Advanced Study in its early years, in term ...

Including:

• Hermann Weyl - Early life and interests
• Hermann Weyl - Geometric foundations of manifolds and physics
• Hermann Weyl - Foundations of mathematics
• Hermann Weyl - Mathematics of relativity
• Hermann Weyl - Topological groups Lie groups and representation theory
• Hermann Weyl - Harmonic analysis and analytic number theory
• Hermann Weyl - Later career
• Hermann Weyl - Personality
• Hermann Weyl - Quotes

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A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes "straight curves"). In Euclidean geometry, exactly one line can be found that passes through any two points. The line provides the shortest connection between the points. Three or more points that lie on the same line are called collinear. Two different lines can either be parallel and never meet, or may intersect at one and only one point. Two planes intersect in at most one line). Lines in a Cartesian plane can be describe ...

Including:

• Line mathematics - Line segment
• Line mathematics - Ray

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In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. A cross-cap that has been closed up by gluing a disc to its boundary is called a real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional nonorientable manifolds are spheres with some ...

Including:

• Cross-cap - Cross-cap model of the real projective plane

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In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. 3-sphere - Explanation. An ordinary sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, R3. A 3-sphere consists of all points equidistant from a single point in R4. Whereas a 2-sphere is a smooth 2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold. In an entirely analogous manner one c ...

Including:

• 3-sphere - Explanation
• 3-sphere - Definition
• 3-sphere - Elementary properties
• 3-sphere - Topological construction
• 3-sphere - Unslicing
• 3-sphere - Unpuncturing
• 3-sphere - Topological properties
• 3-sphere - Coordinate systems on the 3-sphere
• 3-sphere - Hyperspherical coordinates
• 3-sphere - Hopf coordinates
• 3-sphere - Stereographic coordinates
• 3-sphere - Group structure
• 3-sphere - Tangents
• 3-sphere - In literature
• 3-sphere - Related topics
• 3-sphere - External link

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component). Many fields of mathematics include a formally defined property known as connectedness. In each field, the property may be defined differently. How ...

Including:

• Connectedness - Other notions of connectedness
• Connectedness - Connectivity

Read more here: » Connectedness: Encyclopedia - Connectedness

The prototypical example is the ring Z of all integers. Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields. Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomi ...

See also:

Integral domain, Integral domain - Examples, Integral domain - Divisibility prime and irreducible elements, Integral domain - Field of fractions, Integral domain - Characteristic and homomorphisms

Read more here: » Integral domain: Encyclopedia II - Integral domain - Examples

Note the similarity between the definitions of compact and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite". Both of these changes are significant: if we take the above definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases. A hereidtarily paracompact space is a space such that every subspace of it is paracompact. This is equivalent to requiring that every open subspace be p ...

See also:

Paracompact space, Paracompact space - Definitions of relevant terms, Paracompact space - Examples and counterexamples, Paracompact space - Properties, Paracompact space - Partitions of unity, Paracompact space - Variations, Paracompact space - Similarities with compactness, Paracompact space - Product related properties

Read more here: » Paracompact space: Encyclopedia II - Paracompact space - Definitions of relevant terms

Most nylons are condensation copolymers formed by reacting equal parts of a diamine and a dicarboxylic acid, so that peptide bonds form at both ends of each monomer in a process analogous to polypeptide biopolymers. The numerical suffix specifies the numbers of carbons donated by the monomers; the diamine first and the diacid second. The most common variant is nylon 6,6, also called nylon 66, which refers to the fact that the diamine (hexamethylene diamine) and the diacid (adipic acid) each donate 6 carbons to the polymer chain. As with othe ...

See also:

Nylon, Nylon - Chemistry, Nylon - Bulk properties, Nylon - Historical uses, Nylon - Etymology, Nylon - Uses

Read more here: » Nylon: Encyclopedia II - Nylon - Chemistry

Fundamental theorem of calculus - Part I. It is given that Let there be two numbers x1 and x1 + Δx in [a, b]. So we have and . Subtracting the two equations gives . It can be shown that . (The sum of the areas of two adjacent regions is equal to the area of both regions combined.) Manipulating this equation gives . Substit ...

See also:

Fundamental theorem of calculus, Fundamental theorem of calculus - Intuition, Fundamental theorem of calculus - Formal statements, Fundamental theorem of calculus - Corollary, Fundamental theorem of calculus - Proof, Fundamental theorem of calculus - Part I, Fundamental theorem of calculus - Part II, Fundamental theorem of calculus - Examples, Fundamental theorem of calculus - Generalizations

Read more here: » Fundamental theorem of calculus: Encyclopedia II - Fundamental theorem of calculus - Proof