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The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes. [5] Stokes' law is important for understanding the swimming of microorganisms and sperm; also, the sedimentation of small particles and organisms in water, under the force of gravity. [5]
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 vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
In this notation, Stokes' theorem reads as = . In finite element analysis, the first stage is often the approximation of the domain of interest by a triangulation, T. For example, a curve would be approximated as a union of straight line segments; a surface would be approximated by a union of triangles, whose edges are straight line segments ...
In fluid mechanics, non-dimensionalization of the Navier–Stokes equations is the conversion of the Navier–Stokes equation to a nondimensional form. This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain ...
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
Stokes' formula can refer to: Stokes' law for friction force in a viscous fluid. Stokes' law (sound attenuation) law describing attenuation of sound in Newtonian liquids. Stokes' theorem on the integration of differential forms. Stokes' formula (gravity) a formula in geodesy
In acoustics, Stokes's law of sound attenuation is a formula for the attenuation of sound in a Newtonian fluid, such as water or air, due to the fluid's viscosity.It states that the amplitude of a plane wave decreases exponentially with distance traveled, at a rate α given by = where η is the dynamic viscosity coefficient of the fluid, ω is the sound's angular frequency, ρ is the fluid ...