词条 | Parabolic cylindrical coordinates |
释义 |
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges. Basic definitionThe parabolic cylindrical coordinates {{math|(σ, τ, z)}} are defined in terms of the Cartesian coordinates {{math|(x, y, z)}} by: The surfaces of constant {{math|σ}} form confocal parabolic cylinders that open towards {{math|+y}}, whereas the surfaces of constant {{math|τ}} form confocal parabolic cylinders that open in the opposite direction, i.e., towards {{math|−y}}. The foci of all these parabolic cylinders are located along the line defined by {{math|x {{=}} y {{=}} 0}}. The radius {{math|r}} has a simple formula as well that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article. Scale factorsThe scale factors for the parabolic cylindrical coordinates {{math|σ}} and {{math|τ}} are: Differential elementsThe infinitesimal element of volume is The differential displacement is given by: The differential normal area is given by: DelLet {{math|f}} be a scalar field. The gradient is given by The Laplacian is given by Let {{math|A}} be a vector field of the form: The divergence is given by The curl is given by Other differential operators can be expressed in the coordinates {{math|(σ, τ)}} by substituting the scale factors into the general formulae found in orthogonal coordinates. Relationship to other coordinate systemsRelationship to cylindrical coordinates {{math|(ρ, φ, z)}}: Parabolic unit vectors expressed in terms of Cartesian unit vectors: Parabolic cylinder harmonicsSince all of the surfaces of constant {{math|σ}}, {{math|τ}} and {{math|z}} are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written: and Laplace's equation, divided by {{math|V}}, is written: Since the {{math|Z}} equation is separate from the rest, we may write where {{math|m}} is constant. {{math|Z(z)}} has the solution: Substituting {{math|−m2}} for , Laplace's equation may now be written: We may now separate the {{math|S}} and {{math|T}} functions and introduce another constant {{math|n2}} to obtain: The solutions to these equations are the parabolic cylinder functions The parabolic cylinder harmonics for {{math|(m, n)}} are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written: ApplicationsThe classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate. See also
Bibliography
External links
1 : Coordinate systems |
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