Worksheets
This place will become a commented index of
MAPLE worksheets developed, used, or referenced
in class.
While the focus of this class is on theory, there are
nonetheless many places where it is appropriate to use
a computer algebra system for visualization, for
experimentation and quickly calculating some nontrivial
examples.
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ftp-directory,
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egregium.mws: A formal proof (by involved, but straightforward,
calculation) of Gauss' Theorema Egregium.
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The following three are older worksheets from a class that explored the use of MAPLE to
calculate and visualize objects and their properties. But even for our
class they should serve well to develop some intuitive understanding of
some classical objects.
euler.mws: Sectional curvature for a surface.
gauss.mws: Calculate the Gauss curvature and visualziae it w/ color.
gaussmap.mws: The Weingarten map or Gauss map.
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parallel.mws: Parallel transport on 2-dim R-manifolds.
Set up and solve DE. Visualize as still-images or as animation,
in coord plane or on imbedded surface (graph or parameterized).
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connexpolar.mws:
Christoffel symbols for polar /spherical coordinates, picture.
MS.doc picture and comments
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connex1.mws:
Explorations on the graph of a function z=f(x,y): The relation
of the Christoffel symbols to the derivatives of the metric and
first steps towards a covariant derivative of vector fields.
(Includes careful discussion of ambiguities of imbedded surfaces
and "projecting out normal directions".)
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geocurv.mws:
Geodesics on graphs of functions z=f(x,y): Compare geodesic eqn,
curves whose acceleration is normal to the graph, and solving
a constrained minimization problem via Lagrange multipliers.
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geodesics.mws:
Calculate the Riemannin metric, Christoffel symbols, set-up and solve
the geodesic equations. Visualize the results in a number of ways,
in the coordiante plane or on imbedded surfaces. Animations of the
geodesic flow and geodesic spheres. Great images combining geodesics
with color-coding by Gaussian curvature.
The worksheet has been saved with all output as some of it takes time
to recalculate -- however, this results in a 4MBplus file. A small
file without output is
geodesics0.mws.
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viewRmetric.mws:
Calculate the Riemannian metric for imbedded surfaces in R3
and visualize 2-D Riemannain manifolds (brandnew pictures!)
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doublepend.mws: Example from mechancis demonstrating origin
of the metric as inertia tensor.
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riemann1.mws,
riemann1.html:
Demonstration of calculations of Riemannian metric, by hand w/ the tensor package.
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coords1st2nd.mws,
coords1st2nd.html:
Demonstrations of changes to coordinates of the first kind,
and to coordinates of the second kind, respectively.
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hopf.mws,
hopf.html:
Explorations of the Hopf map: S3->S2.
Local coord's via stereographic projection.
rank?
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ellipse.mws,
ellipse.html:
An advanced level intro (meant for a presentation w/ guided discussion)
to the capabilitiues of MAPLE.
Example: Arclength and curvature of an ellipse.
Invariance of plane-curvature under rotations.
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frenet.mws:
(.html/.gif):
Integrating the Frenet equations -- completely rewritten Feb.3
Until this reference page starts to take form, please
consult similar pages for recent classes: