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An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus. [3 ...
In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. [1] It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations. [2]
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in ). In one dimension, it is equivalent to the fundamental theorem of calculus.
Because the exterior derivative d has the property that d 2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The k -th de Rham cohomology (group) is the vector space of closed k -forms modulo the exact k -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces ...
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes [1] (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based ...
Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller . In this technique, the effect of a particular optical element is represented by a Mueller matrix—a 4×4 matrix that is an overlapping generalization of the Jones matrix .
The Stokes I, Q, U and V parameters. The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation.They were defined by George Gabriel Stokes in 1851, [1] [2] as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of ...