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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 ...
Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem, [3] uses Green's second identity to derive the solution of the homogeneous scalar wave equation at an arbitrary spatial position P in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface as the boundary of some volume including P.
The Kirchhoff–Helmholtz integral combines the Helmholtz equation with the Kirchhoff integral theorem [1] to produce a method applicable to acoustics, [2] seismology [3] and other disciplines involving wave propagation.
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid. = + + +, = + +, = (~ +) = ^, = ^ where and are the angular and linear velocity vectors at the point , respectively; ~ is the moment of inertia tensor, is the body's mass; ^ is a unit normal vector to the surface of the body at the point ; is a pressure at this point ...
Kirchhoff showed that in many cases, the theorem can be approximated to a simpler form that is equivalent to the formation of Fresnel's formulation. [3] For an aperture illumination consisting of a single expanding spherical wave, if the radius of the curvature of the wave is sufficiently large, Kirchhoff gave the following expression for K(χ ...
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.
Kirchhoff's laws, named after Gustav Kirchhoff, may refer to: Kirchhoff's circuit laws in electrical engineering; Kirchhoff's law of thermal radiation; Kirchhoff equations in fluid dynamics; Kirchhoff's three laws of spectroscopy; Kirchhoff's law of thermochemistry; Kirchhoff's theorem about the number of spanning trees in a graph
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
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