Stokes' theorem

Stokes' theorem equates the surface integral of a vector field’s curl over an oriented surface with the circulation of the field around its boundary curve.

Curl: A differential operator measuring the local rotation of a vector field. Stokes' theorem integrates this local rotation across the surface.
Vector field: A function assigning a vector to each point in a space. The theorem applies to vector fields whose curl is integrated over a surface.
Ampère's circuital law: A law relating magnetic fields around a closed path to the electric current passing through the path. Stokes' theorem converts its circulation form into a surface integral of curl.
Divergence theorem: A theorem equating outward flux through a closed surface with the volume integral of divergence inside it. It relates a different differential operator and boundary dimension than Stokes' theorem.
Line integral: An integral that accumulates a function or vector field along a curve. The theorem's boundary term is a line integral of the field.
Differential form: A mathematical object that can be integrated over oriented curves, surfaces, and higher-dimensional spaces. Differential forms express Stokes' theorem in a coordinate-independent language.
Faraday's law of induction: A law stating that changing magnetic flux induces an electromotive force around a closed loop. Stokes' theorem relates the loop integral of electric field to magnetic-field change across a surface.
Green's identities: Integral identities relating functions and their derivatives over a region to integrals over its boundary. They connect boundary and interior integrals but involve scalar differential operators.
Surface integral: An integral that accumulates a scalar or vector quantity over a surface. The theorem integrates curl across an oriented surface.
Boundary (topology): The edge of a geometric or topological space, defined so that it separates the space from its exterior. The curve in the theorem is precisely the boundary of the chosen surface.

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