 About 20,900 results  books.google.com 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
Problem ... 

 books.google.com The Helmholtz differential equation can be solved by SEPARATION OF
VARIABLES in only 11 coordinate systems, 10 of which ... See also LAPLACE'S
EQUATIONBIPOLAR COORDINATES Helmholtz Differential Equation —
Bispherical ... 

 books.google.com 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 ... 

 books.google.com (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
−q=0. 

 books.google.com 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. Twocenter bipolar coordinates are
... 

 books.google.com 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
... 

 books.google.com [14] 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
particles: ... 

 books.google.com 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, ... 

 books.google.com TABLE 21 Twodimensional 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, ... 

 books.google.com ... 439) INTRODUCTION Because only steadystate 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
coordinates. 

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