LECTURE: MATH 161, SPRING 2017, M-W-F: 11-11:50 @ ICF 103
INSTRUCTOR: NAM TRANG
OFFICE HOURS: M 12--1, W 1:30--2:30 @ RH 410N

TA: AGRESS, D.
DISCUSSION: T-Th: 8-8:50 @ MPAA 330


COURSE POLICIES     COURSE SYLLABUS     UCI Academic Honesty Policies     UCI Student Resources

HOMEWORK ASSIGNMENTS:     HW1     HW1 Solution (part 1)     HW1 Solution (part 2)     HW2     HW2 Solution     HW3     HW3 Solution    HW4    HW4 Solution    HW5    HW5 Solution    HW6    HW6 Solution

FURTHER READING/PROBLEMS: Read pages 65 and 66 (related to Pasch's axiom) and do problems 2.2.4-2.2.12.
Extra problems on complex numbers and vectors pdf ; Solutions: pdf.
Extra problems on isometries and hyperbolic geometry pdf ;   Solutions: pdf

MIDTERM/FINAL REVIEWS AND SAMPLE FINAL: Sample Final

FINAL'S WEEK:

QUIZZES:     Quiz 1: Thu Apr 13, in discussion; material: topics in lecture covered in Homework 1
                        Quiz 2: Tue Apr 25, in discussion; material: topics in lecture covered in Homework 2
                        Quiz 3: Thu May 4, in discussion; material: topics in lecture covered in Homework 3 (sections on vectors and angles, including dot products and its applications (triangle inequality and Cauchy-Schwarz inequality))
                        Quiz 4: Thu May 18, in discussion; material: topics in lecture covered in Homework 4 (minus problem 4)


COURSE PROGRESS

Week 1:
M: Went over class policies, syllabus; some history/context for studying geometry; defined axiomatic systems and gave examples (chess, semi-group).
W: Continued with axiomatic systems; defined consistency, independence and completeness of axiomatic systems; defined models of an axiomatic system (e.g. (Z,+) is a model of semi-group/group axioms); stated Godel's incompleteness theorem; discussed the axioms A1--A4 in Section 1.4.
F: Introduced Euclid's axioms/postulates (A1-A5 and P1-P5) for geometry and proved I.1.

Week 2:
M: Proved I.16, I.27 (headed towards proving P5 is equivalent to Playfair's postulate (modulo the other postulates))
W: Proved I.28, I.29, I.32 (consequence: sum of three angles in a triangle is 2 right angles); proved Playfair's postulate is equivalent to Euclid's fifth postulate (modulo the other postulates/axioms).
F: Discussed congruent triangles (SSS, SAS, ASA criteria); explained why (SSA and AAA are not criteria for determining if two triangles are congruent).

Week 3:
M: Discuss Pasch's Axiom; defined parallelogram, rectangle, defined area of a rectangle.
W: Computed area of a parallelogram and a triangle; proved Pythagorean theorem.
F: Define cevians; proved Ceva's theorem; defined similar triangles; application of similar triangles: measure height of a building; stated the AAA criterion for similar triangles.

Week 4:
M: Proved the AAA criterion for similar triangles;
W: Proved inscribed angle is half the center angle with the same arc (in a circle); corollary: inscribed angles in a semicircle is a right angle and two inscribed angles with the same arc are congruent; defined tangent to a circle and showed that tangency is equivalent to being perpendicular to the line connecting the center of the circle with the point of tangency.
F: Proved the existence of tangents; introduced analytic geometry:coordinate system, vectors, vector operations: addition, scalar multiplication, dot product, length of a vector.

Week 5:
M: vector equation of a line through 2 points A, B; proved a couple of geometric theorems using vectors and coordinates in combination with pure geometry: proved OACB is a parallelogram where A=(a,b), B=(c,d), and C=(a+c,b+d); AG = 2/3 AD etc.; proved the law of cosines and law of sines
W: Proved v.w = |v||w| cos\Theta and as a corollary, get the Cauchy-Schwarz inequality: |v.w| <= |v||w| and presented a simple application of Cauchy-Schwarz. Introduced complex numbers: operations on complex numbers; representations of complex numbers: vector form, polar form, and Euler's formula.
F: Defined the point at infinity z_inf on the complex plane (Riemann sphere = the complex plane + {z_\inf}); described stereographic projection and computed coordinates of points on the unit sphere via the stereographic projection; (proof: Page 1     Page 2 )

Week 6:
M: Midterm
W: Proved stereographic projection maps circles on the unit sphere to circles or lines in the complex plane ( Page 1     Page 2 )
F: Defined transformations as 1-1 and onto maps; defined isometries as length-preserving transformations. Showed that isometries preserve angles, map lines to lines, map parallel lines to parallel lines, preserves betweenness. Proved that if an isometry fixes two points A,B then it fixes the whole line through A,B.

Week 7:
M: Proved if an isometry fixes 3 noncollinear points, it fixes all points; defined central isometries, showed central isometries are linear maps and preserve dot products; showed that every isometry is either a central isometry or a central isometry followed by a translation.
W: Review basic linear algebra: orthogonal matrix, orthonormal bases, matrix that represents a linear transformation; classify central isometries: these are either rotation (around the origin) or reflection across a line through the origin; computed the matrix for \rho_\theta, the rotation by angle \theta.
F: Gave examples of computing eigenvalues and eigenvectors of central isometries; showed that every central isometry is either a rotation around the origin or a reflection across a line going through the origin; discussed what happens if one composes two central isometries. (lecture notes: Page 1     Page 2     Page 3 )

Week 8:
M & W: Showed rotation followed by a translation is a rotation (around some point x_fix); showed how to compute the coordinates of x_fix; showed reflection across a line through the origin followed by a translation is either a reflection (across a line parallel to the original line) or a glide. (lecture notes: Page 1     Page 2     Page 3 )
F: showed f T_a = T_b f for some vector b, where T_a is the translation by vector a, f: a central isometry. Worked out the example of computing the composition of: reflection about the line y=x followed by gliding along the y-axis upward by 1 unit, followed by rotation about the point (1,1) by 90-degrees

Week 9:
M: Holiday. No class.
W: Introduced hyperbolic geometry; Defined the Poincare model; Defined distance function in the Poincare model.
F: Worked through an example of computing hyperbolic distance between two points in the Poincare model; discussed the relationship between hyperbolic distance and Euclidean distance; proved that the Poincare disk satisfies P1-P4 and P5^*.

Week 10:
M: Proved the fundamental theorem of limit parallels; gave an example of computing limit parallels in the Poincare model.
W: Continued with the example of computing limit parallels in the Poincare model; Defined Saccheri and Lambert quadrilaterals; stated some basic facts about them.
F: Proved that the sum of interior angles of a triangle in hyperbolic geometry is < 2 right angles; proved that if two triangles are similar, they must be congruent.