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Coordinate system changing

Bipolar cylindrical coordinate system on 3-space

What we do?

Find metric, connection and Laplace operator on 3-space in bipolar cylindrical coordinate system:
x = a*sinh(v)/(cosh(v)-cos(u))
y = a*sin(u)/(cosh(v)-cos(u))
z = w

Solution:

Domain[manifold]manifold - string - a manifold name or a name of a manifold domain.
Metric[id→expr]id - variable - metric identifier, expr - expression - metric declaration.
Connection[id]id - variable - connection identifier.

Necessary functions.

Load the Atlas package:
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Space

First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. To define the space we declare domain, forms, vectors, coframe, frame, flat metric and calculate connection (it is equal to zero of course).
Declare domain:
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Declare some forms:
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Declare some vectors:
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Declare coframe:
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Declare frame:
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Declare a flat metric:
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Calculate the connection of the metric:
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Now the working space is defined completely and we can start to solve the problem.

Bipolar cylindrical

Define new domain:
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Declare 1-form for the domain coframe:
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Declare vectors for the domain frame:
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Declare coframe on the domain:
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Declare frame of the domain:
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Declare mapping of the domain into R3:
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Visualize the mapping:
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Now we can calculate metric induced on the domain by the mapping.
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Calculate connection:
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To calculate Laplace operator one can use grad and div operators.
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