Study Guide for Test 3
There is a lot of notation in this section. You are not
expected to memorize it. Instead,
know the ideas behind the first fundamental form,
metrics, Gauss curvature, Christoffel symbols, and geodesics, and if
you are given the notation, you should recognize what it corresponds to
For example, you should recognize th geodesic equations if I gave them to
you as in Amy Ksir's Geodesic worksheet in the summation notation,
or in the form given on p. 230
of the text, and also understand that they arise from setting the
geodesic curvature (recall the material on the last test) equal to 0,
leaving only normal curvature that is not felt intrinsically by a bug.
I will give you one proof. As we did in class, show how
E, F, and G and the metric equation
arise from our usual definition of arc length along a curve.
You can find this in notes, or in a slightly different version as the first
Review the following Maple worksheets that are available on the class
highlights page, and know the big picture and understand the point of these
1) Applications of the first fundamental form
(Local Isometry: Catenoid and Helicoid, and Area: Sphere,
Strake, Hyperbolic Geometry, Cone.)
2) Maple worksheet on Gauss curvature and the sphere, helicoid, and
For example, I might present some Maple commands and
output and ask what they showed
us. More specifically, I might present the Maple output for the area
of a disk on the cone and ask you to explain where the integral command
came from (The double integral of the sqrt(EG-F^2), and the limits
on the parameters like in class and calc 3) and compare the result to a
covering space argument, like we did in class, or I might ask you to
set up one of these integrations for one of the surfaces.
As another specific example, I might give you the commands that show
that the helicoid and catenoid have the same E, F, and G, and ask you
to explain what this means (ie that they are locally the same - isometric -
and can be deformed into each other like the avi movie I showed), and
also explain how we could have used just this to understand why the
Gauss curvature is the same without calculating it like we did in the
other Maple worksheet
(Root showed us via p. 134-6 in the book that the Gauss curvature depends
only on E, F, and G and its derivatives, even though the definition needs
the extrinsic normal vector to the surface, so if E, F, and G are the same,
the Gauss curvature will be too.) You should also understand intrinsic
Gauss curvature intuition (whether it is zero, positive, or negative).
Go over the geodesic worksheet we completed in class.
Frank created solutions which are up on WebCT.
I would not expect you to memorize these - ie I would give you the formulas,
like they are given in the worksheet,
and might ask you to apply them, like in 1 (b) and (c). This requires
that you understand and can use the summation notation...
I might ask you to find just one of the Christoffel symbols for
the sphere instead of all of them in 2 (c).