Surface geometry in Atlas 2D/3D Wizard

This tutorial illustrates how to use the Atlas 2D/3D Wizard package to solve problems in elementary differential geometry. As an example we find the geometry of the torus.

What we have?

Torus in 3D space (Cartesian coordinate system):
x = (a+b*Cos[u])*Cos[v]
y = (a+b*Cos[u])*Sin[v]
z = b*Sin[v]

What we do?

  • We calculate the following quantities: first fundamental form (surface metric), second fundamental form and field of mean curvature vectors.
    • Besides that we calculate the connection, Christoffel symbols, curvature, curvature tensor field (Riemann tensor field), Ricci tensor field, Gauss curvature and Laplace operator.
    Run Atlas 2D/3D Wizard and press NEXT button:
    Click for copyable input
    Choose 3D- space and press NEXT button:
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    Select RECTANGULAR coordinate system and press NEXT button:
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    Select SURFACE and press NEXT button:
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    Enter the surface equations and press NEXT button (u, v are surface coordinates):
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    Press Check-Out button or just skip this step and press NEXT button:
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    Specify a notebook to save the Mathematica code (Output.nb by default):
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    Press FINISH button and execute in Mathematica the generated file:
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    You will see the results of the execution in the notebook: surface metric, second fundamental form, mean curvature vector, connection, Christoffel symbols, curvature, Riemann and Ricci tensors, Gauss curvature, Laplace operator.
    It takes 2 minutes to solve this problem from the beginning to the end! Just try it.