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Surfaces in R3

Abstract revolutionary surface

Examples:

Find metric and second fundamental form of the following revolutionary surface:

Examples:

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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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.
Just for right simplification:
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Surface

Define the surface as a manifold:
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Declare functions:
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Declare 1-form for surface coframe
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Declare vectors for surface frame:
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Declare coframe on the surface:
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Declare frame of the surface:
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Declare mapping of the surface into R3:
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One can also calculate metric induced on the surface by the mapping.
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Calculate invariants of the mapping:
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Let us extract the mean curvature vector field:
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Let us extract the second fundamental form:
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Now we can calculate the corresponding tensor:
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