Information about Final Exam for Math 4513-006H

Mathematics 4513-001 - Euclidean and non-Euclidean Geometry - Fall 2008

Information about Final Exam

The Final Exam will be in the usual classroom on Friday, December 19, from 9:30-11:45 a. m. Note that this is a change from the scheduled starting time of 8:00 a. m. If you wish to begin (and finish) the exam earlier, please contact me and I will arrange that.

Calculators or other mechanical assistance are not needed and are not to be used. Blank paper will be provided on which to write your solutions, so all you will need is something to write with.

Some of the exam problems will be similar to homework problems, while others will draw upon the material presented in the lectures. As on any exam, it is wise to start with the problems that you feel confident that you know how to do, before moving on to others. If asked for a definition, give the definition that we have used in this course.

The exam has 77 points possible. It will give more weight to the second half of the course. Many of the problems are fairly short and can be done quickly, although I expect that few if any students to be able to do all of the problems.

The following topics are very likely to appear, although the exam is not limited to these topics:

  1. Lines, their description as P + [v], direction of a line, unit normals.
  2. Isometries, their definition and basic properties.
  3. Isomorphisms: checking injectivity, surjectivity, and the homomorphism property.
  4. Subgroups and normal subgroups.
  5. The matrices rot(θ) and ref(θ).
  6. Translations and rotations in the plane, (just the definitions and most basic information).
  7. Three Reflections Theorems (we studied three of them).
  8. Definition of the cross product using the determinant, using the definition to derive basic properties of the cross product.
  9. Lines, poles, and distance on S². The line containing two points. Perpendicular lines.
  10. Isometries of S²: reflections, rotations, and glide-reflections, their definitions and their matrix representations for a well-chosen basis.
  11. The Orthonormal Basis Theorem.

The following topics do not appear, at least not explicitly: Cauchy-Schwarz inequality, Triangle inequality, distance between parallel lines, formulas from the Existence of Perpendicular Lines theorem, finding intersection point of two lines, dilations, orders of elements in groups, cyclic groups, cosets, O(2), SO(2), O(3), SO(3).

There is no need to memorize long proofs of theorems. Familiarity with the homework problems would be a good start on your preparation.

If you want to see some exams I have written in other courses, many of them are available at the links that appear on my course pages page. Some were 50-minute classes, but most were 75-minute classes.

1. Lines, their description as P + [v], direction of a line, unit normals.
2. Isometries, their definition and basic properties.
3. Isomorphisms: checking injectivity, surjectivity, and the homomorphism property.
4. Subgroups and normal subgroups.
5. The matrices rot(θ) and ref(θ).
6. Translations and rotations in the plane, (just the definitions and most basic information).
7. Three Reflections Theorems (we studied three of them).
8. Definition of the cross product using the determinant, using the definition to derive basic properties of the cross product.
9. Lines, poles, and distance on S². The line containing two points. Perpendicular lines.
10. Isometries of S²: reflections, rotations, and glide-reflections, their definitions and their matrix representations for a well-chosen basis.
11. The Orthonormal Basis Theorem.