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|7.12. Associated Legendre Functions, 192. Problems, 199. SPHERICAL |
HARMONICS. APPLICATIONS, Page 204. 8.1. Introductory Remarks, 204. 8.2.
Solution of Laplace's Equation in Spherical Coordinates, 205. 8.3. The Dirichlet
|The Helmholtz differential equation can be solved by SEPARATION OF |
VARIABLES in only 11 coordinate systems, 10 of which ... See also LAPLACE'S
EQUATION-BIPOLAR COORDINATES Helmholtz Differential Equation —
|Laplace's equation can be written in terms of the new variables 4 The surface £ = |
const is a prolate spheroid with interfocal ... /17.6 Bipolar coordinates (£, 77, -)
Bipolar coordinates are useful for the analysis of liquid sloshing in a horizontal ...
|(5.87) 2 cosh τ +cosσ 2 coshτ +cosσ ,y= The zero order solution satisfying the |
Laplace equation ∇2θ(0) =0and the ... The inhomogeneity in the first order
equation written in the bipolar coordinates turns out to be constant: θ(1)ττ+θ(1)σσ
|birational transformation The scale factors are cosh v - cos u a cosh u - cos « The |
Laplacian in bipolar coordinates is 2 (cosh v - cos «)2 / 32 32 9 I ^ 9^ ' "n "5 a.1
Vdu.1 Sir y - Laplace's equation is separable. Two-center bipolar coordinates are
|For problems connected with two spheres bipolar coordinates may be used; the |
appropriate substitution is'l' a sin \[I' a sinh ... ..(246) satisfies Laplace's equation
and that when we use bipolar coordinates the corresponding solution is ' (cosh a-
| is the analytical approach to solving the Laplace equation: instead of using |
the approximation about a mere ... Second, an analytical solution of Laplace
equation in bipolar coordinates was obtained for the case of two dissimilar
|Apparently only the LAPLACE equation is separable in toroidal coordinates. ... Q |
satisfy the LEGENDRE equation (5.1}0) Mathematica uses the toroidal
coordinate system u,w,p/ti,a, which is obtained by rotating bipolar coordinates u,
v, 2, ...
|TABLE 21 Two-dimensional Laplace operator in some curvilinear orthogonal |
systems of coordinates Coordinates ... cosh cos, = sinh sin 0 ≤ < , 0 ≤ < 2 1 2(
sinh2+sin2 ) 2 2 + 2 2 Bipolar coordinates , = sinhcosh−cos, = sin cosh −cos 0 ≤
< 2, ...
|... 439) INTRODUCTION Because only steady-state temperature problems in |
polar coordinates are considered in this section, the first thing we must do is
convert the familiar Laplace's equation in rectangular coordinates to polar
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