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The de Rham theorem establishes the isomorphism between the de Rham cohomology of a smooth manifold and the singular cohomology with real coefficients. This deep result bridges the purely algebraic and differential geometric approaches to topology.

Stokes' Theorem generalizes several theorems from vector calculus, including the Fundamental Theorem of Calculus. In the context of de Rham cohomology, it relates the integral of a differential form over the boundary of a manifold to the integral of its exterior derivative over the manifold itself.

In de Rham cohomology, the cohomology group is a vector space formed by the quotient of closed forms modulo exact forms. It measures the failure of the Poincaré lemma to hold globally and provides algebraic invariants of the manifold.

De Rham cohomology is a mathematical tool in differential geometry and algebraic topology that uses differential forms to study topological properties of manifolds. It shows the relationship between differential forms, smooth functions, and the topology of the manifold.

A closed form is a differential form whose exterior derivative is zero. In de Rham cohomology, closed forms represent potential cohomology classes, encapsulating important topological information.

An exact form is a differential form that is the exterior derivative of another form. In the context of de Rham cohomology, exact forms are considered trivial since they do not contribute to the cohomology groups of the manifold.

The exterior derivative is an operator that takes a k-form to a (k+1)-form. It generalizes the concept of taking the derivative of a function and is a crucial component in the definition of de Rham cohomology.

The cup product endows the cohomology ring with a graded algebraic structure. In de Rham cohomology, it combines differential forms using the wedge product and provides information about the intersection properties of submanifolds and other geometric interactions.

A differential form is an object in differential geometry that generalizes the concept of functions and vector fields. It can be integrated over a manifold and is key in defining integrals in higher dimensions.