Difference between revisions of "Bicylindrical coordinates"
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| − | + | The numbers $ \tau , \sigma $ | |
| + | and $ z $ | ||
| + | related to the rectangular Cartesian coordinates $ x, y $ | ||
| + | and $ z $ | ||
| + | by the formulas | ||
| + | |||
| + | $$ | ||
| + | x = | ||
| + | \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } | ||
| + | ,\ \ | ||
| + | y = | ||
| + | \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } | ||
| + | ,\ \ | ||
| + | z = z, | ||
| + | $$ | ||
| + | |||
| + | where $ 0 \leq \sigma < \pi , -\infty < \tau < \infty $. | ||
| + | The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $ \tau = \textrm{ const } $), | ||
| + | the family of circular cylinders orthogonal to the former ( $ \sigma = \textrm{ const } $), | ||
| + | and the planes ( $ z = \textrm{ const } $). | ||
| + | The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $ xy $- | ||
| + | plane parallel to the $ z $- | ||
| + | axis. | ||
The Lamé coefficients (cf. [[Lamé coefficients|Lamé coefficients]]) are: | The Lamé coefficients (cf. [[Lamé coefficients|Lamé coefficients]]) are: | ||
| − | + | $$ | |
| + | L _ \sigma = L _ \tau = \ | ||
| + | |||
| + | \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } | ||
| + | ,\ \ | ||
| + | L _ {z} = 1. | ||
| + | $$ | ||
The Laplace operator is: | The Laplace operator is: | ||
| − | + | $$ | |
| + | \Delta f = | ||
| + | \frac{1}{a ^ {2} } | ||
| + | |||
| + | ( \cosh \tau - \cos \sigma ) ^ {2} | ||
| + | \left ( | ||
| + | |||
| + | \frac{\partial ^ {2} f }{\partial \sigma ^ {2} } | ||
| + | + | ||
| + | |||
| + | \frac{\partial ^ {2} f }{\partial \tau ^ {2} } | ||
| + | \ | ||
| + | \right ) + | ||
| + | |||
| + | \frac{\partial ^ {2} f }{\partial z ^ {2} } | ||
| + | . | ||
| + | $$ | ||
Latest revision as of 10:59, 29 May 2020
The numbers $ \tau , \sigma $
and $ z $
related to the rectangular Cartesian coordinates $ x, y $
and $ z $
by the formulas
$$ x = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,\ \ y = \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } ,\ \ z = z, $$
where $ 0 \leq \sigma < \pi , -\infty < \tau < \infty $. The coordinate surfaces are: the family of pairs of circular cylinders with parallel axes ( $ \tau = \textrm{ const } $), the family of circular cylinders orthogonal to the former ( $ \sigma = \textrm{ const } $), and the planes ( $ z = \textrm{ const } $). The system of bicylindrical coordinates is obtained as the result of translation of the system of bipolar coordinates in the $ xy $- plane parallel to the $ z $- axis.
The Lamé coefficients (cf. Lamé coefficients) are:
$$ L _ \sigma = L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } ,\ \ L _ {z} = 1. $$
The Laplace operator is:
$$ \Delta f = \frac{1}{a ^ {2} } ( \cosh \tau - \cos \sigma ) ^ {2} \left ( \frac{\partial ^ {2} f }{\partial \sigma ^ {2} } + \frac{\partial ^ {2} f }{\partial \tau ^ {2} } \ \right ) + \frac{\partial ^ {2} f }{\partial z ^ {2} } . $$
Bicylindrical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicylindrical_coordinates&oldid=16217