Generalized Stokes theorem

The generalized Stokes theorem equates the integral of a differential form’s exterior derivative over an oriented manifold with the form’s integral over its oriented boundary.

Exterior derivative: An operation that maps differential forms to forms of one higher degree and satisfies a graded product rule. Applying it to the form on the manifold produces the integrand on the left side.
Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate charts. The theorem is formulated on manifolds where differential forms and integration are defined.
Fundamental theorem of calculus: The theorem relating a function’s derivative integrated over an interval to its endpoint values. It is the one-dimensional instance of integrating a derivative over a manifold with boundary.
Augustin-Louis Cauchy: A French mathematician whose work shaped analysis and complex function theory. Cauchy’s integral theorem is an early influential instance of integration constrained by boundary structure.
Differential form: A smooth field of alternating multilinear functions that can be integrated over oriented manifolds. The theorem applies to forms of degree one less than the manifold’s dimension.
Manifold with boundary: A smooth manifold whose coordinate neighborhoods may include points on a boundary. It supplies the domain and boundary appearing in the theorem.
Green's theorem: A plane theorem relating a line integral around a simple closed curve to a double integral over its region. It follows by applying generalized Stokes to a two-dimensional region.
Bernhard Riemann: A German mathematician whose work transformed geometry, analysis, and the theory of complex functions. Riemannian integration and manifold ideas helped set the geometric stage for the theorem.
Orientation: A consistent choice of orientation for the tangent spaces of a manifold. Compatible orientations determine the boundary integral’s sign.
Chain complex: A sequence of algebraic objects linked by boundary maps whose consecutive compositions vanish. The identity that the boundary of a boundary vanishes underlies the theorem’s boundary structure.

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