Search results
Results From The WOW.Com Content Network
Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in ...
The integral has the following form for a monochromatic wave: [2] [3] [4] = [^ ^],where the integration is performed over an arbitrary closed surface S enclosing the observation point , in is the wavenumber, in is the distance from an (infinitesimally small) integral surface element to the point , is the spatial part of the solution of the homogeneous scalar wave equation (i.e., (,) = as the ...
The Crofton formula generalizes to any Riemannian surface or more generally to two-dimensional Finsler manifolds; the integral is then performed with the natural measure on the space of geodesics. [2] More general forms exist, such as the kinematic formula of Chern. [3]
The definition of surface integral relies on splitting the surface into small surface elements. A surface integral generalizes double integrals to integration over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral. The function to be integrated may be a scalar field or a ...
A geometrical arrangement used in deriving the Kirchhoff's diffraction formula. The area designated by A 1 is the aperture (opening), the areas marked by A 2 are opaque areas, and A 3 is the hemisphere as a part of the closed integral surface (consisted of the areas A 1, A 2, and A 3) for the Kirchhoff's integral theorem.
The calculus of moving surfaces [1] provides analogous formulas for volume integrals over Euclidean domains, and surface integrals over differential geometry of surfaces, curved surfaces, including integrals over curved surfaces with moving contour boundaries.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell ...