Test 2: Surfaces

It is time for our second exam in order to be sure that everyone reviews some of the fundamental concepts before we move on to the geometry of space-time and applications to general relativity

At the Exam
  • You may make yourself some reference notes on the very small card I hand out. The mini reference card must be handwritten. Think of the card as a way to include some important concepts, computations, or derivations that you aren't as comfortable with. You won't have room for much, so you should try to internalize as much as you can.
  • One scientific calculator or graphing calculator allowed (but no cell phone nor other calculators bundled in combination with additional technologies). I don't see that you would need this, but I know some people like to have it with them.
  • You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be stand alone--no cell phone, internet or other technological connections)
  • Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.

    There will be three parts to the exam.
    Part 1: Fill in the blank/short answer
    Part 2: Calculations and Interpretations
    Part 3: Short Derivations/Proofs

    I suggest that you review your class notes, the class highlights page, including clicker questions, and go over ASULearn solutions to the projects, and the surfaces glossary.

    Part 1: Fill in the blank/short answer There will be short answer questions, such as providing:

  • definitions and big picture ideas related to any of the topics in the glossary on surfaces including the parametrization, shape and geometric properties of the 9 surfaces listed there (review class notes, homework and clickers for those).
  • questions similar to previous clicker questions (see the class highlights page), where you fill in a blank rather than answer as a multiple choice.
  • other questions on material from class and homework solutions

    Note: there is often more than one answer possible for fill in the blank questions: choose one response. Full credit responses demonstrate deep understanding of differential geometry. Informal responses are fine as long as they are correct.

    Part 2: Calculations and Interpretations There will be some by-hand computations and interpretations
  • Finding Xu and Xv for a surface
  • Finding the curvature vector dT/ds = T'(t)/speed of a curve on a surface, a normal to a surface Xu x Xv, the projection of the curvature onto the normal (the normal curvature), and the geodesic curvature vector (what is left over, if anything, for the bug to feel) for a curve. Especially review the plane and the cylinder (where the computations are quicker than they would be on other surfaces).
  • Interpreting whether curves are geodesics via a given geodesic curvature, a covering argument, as well as geometric/physical or other arguments relating to whether the curvature vector is completely in the normal direction or not. [For example, for a circle on a surface, we know the curvature vector of any circle points in to the center of the circle. Combine this with intuition about the normal to a surface to say whether the curvature vector is parallel to the normal and hence gives a geodesic, or not]
  • Finding E, F and G and interpreting whether the Pythagorean theorem holds (E=G=1, F=0) or not, or whether Xu and Xv are perpendicular (F=0).
  • Computing I, II, K, and the shape operator on the plane or the cylinder (where the computations are quicker than they would be on other surfaces)
  • Interpreting selections from Maple files (where I would present you with some code and/or Maple output):
    Maple file on geodesic and normal curvatures
    Maple file on Applications of the first fundamental form (isometries and area)
    Maple file on strake and I, II and K

    Part 3: Short Derivations/Proofs There will be some short proofs - the same as we've seen before. Review the following:

  • How E, F, and G and the metric equation arise from our usual definition of arc length along a curve, as on class slides
  • A geodesic must be a constant speed curve as on class slides
  • How the determinant of the metric form gives the area of a flat Xu, Xv parallelogram as on class slides
  • Show how l was derived from class on 3/22
  • In hyperbolic geometry models with radius r as the interior radius of the annuli of width delta, with delta approaching 0, prove that if two geodesics are d units apart along the base curve and we travel c units away from the base curve along each geodesic, then they are at a distance of d exp(-c/r). We filled in the details, after starting from class slides