Closed and exact differential forms & Cup product - Unionpedia, the concept map

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Difference between Closed and exact differential forms and Cup product

Closed and exact differential forms vs. Cup product

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero, and an exact form is a differential form, α, that is the exterior derivative of another differential form β. In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex.

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De Rham cohomology

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

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Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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