Coordinate system changing

Parabolic coordinate system on a plane

What we do?

Find metric, connection and Laplace operator on a plane in parabolic coordinate systems:
x = (u^2-v^2)/2
y = u*v

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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Plane

First of all we have to describe the space we are working in. The space is 2-dimensional Euclidean (flat) space i.e. a plane. 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.

Parabolic

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 R2:
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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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