McMASTER UNIVERSITY
MATH 3B3 MIDTERM TEST
Dr. M. Wang
DATE: October 27, 2009.
DURATION OF TEST: 90 minutes
FAMILY NAME: FIRST NAME:
STUDENT NUMBER: _ TUTORIAL NUMBER:
This test paper contains 3 questions on a total of 10 pages. You are responsible
for ens
Math 3B3 Winter 2004
Solutions to Assignment 5
1. What kind of a number is the area of a P-triangle? Can an asymptotic or a doubly asymptotic
P-triangle have area equal to ? Explain your answer!
Solution. Since the area of any P-triangle is equal to its d
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MATHEMATICS 3B03 TEST 1
Day Class Dr. A. Nicas
Duration of Examination: 50 minutes
McMaster University Midterm Examination October 5, 2011
NAME: 50 Qu'i 1 (MS STUDENT No.2
THIS TEST PAPER INCLUDES 6 PAGES AND 4 QUESTIONS. YOU ARE RESPON-
SIBLE FOR ENSURIN
MATHEMATICS 3B03 TEST 2
Day Class Dr. A. Nicas
Duration of Examination: 50 minutes
McMaster University Midterm Examination November 9, 2011
NAME: Q Mk ] 0713 STUDENT No.2
THIS TEST PAPER INCLUDES 6 PAGES AND 4 QUESTIONS. YOU ARE RESPON
SIBLE FOR ENSURING
Question A. For each of the following statements circle T if it is true and F if it is false.
[10 marks]
@ F (1) A regular curve with zero curvature is part of a straight line.
T G) (2) The curve 7(t) : (cos3(t), sin3(t) with 00 < t < +00 is a regular cur
MATH 3B03
Assignment 5
Due November 28, 2011
1. Calculate the Gaussian and mean curvatures of the parametrized surface
(u, v) = (u + v, u v, uv).
2. Let be a curve on a surface S. Assume that the normal curvature of is zero. Show
that the Gaussian curvatu
MATH 3B03
Assignment 4
Due November 14, 2011
1. Calculate the second fundamental forms of the following parametrized surfaces:
(i) (u, v) = (u cos(v), u sin(v), ln(u) where u > 0 and 0 < v < 2.
(ii) (u, v) = (u cos(v), u sin(v), v).
2. Let U R2 be a non-e
MATH 3B03
Assignment 3
Due October 31, 2011
1. Calculate the rst fundamental forms of the following parametrized surfaces:
(i) (u, v) = (u cos(v), u sin(v), ln(u) where u > 0 and 0 < v < 2.
(ii) (u, v) = (v cos(u), v sin(u), u). This is a helicoid (see Ex
MATH 3B03
Assignment 2
Due October 17, 2011
1. This exercise concerns open sets in R3 (see Pressley, p. 67).
(i) Let Dr (a) and Ds (b) be open balls in R3 (see Pressley, p. 68). Show that the intersection Dr (a) Ds (b) is an open set in R3 . Use this to s
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Math 3B3 Winter 2004
Solutions to Assignment 4
Questions 1 and 2 are statements in Euclidean geometry of circles and spheres.
1. Let i (for i = 1, 2) be a circle with center Oi and radius ri . Suppose 1 and 2 are disjoint
and that neither one is contained
Math 3B3 Winter 2004
Assignment 5
Questions 1-3 are concerned with Hyperbolic area and asymptotic triangles.
1. Read pages 294299 in Coxeters Introduction to Geometry. Sections 16.4 and 16.5 discuss
area in Hyperbolic geometry. Note that Coxeter uses the
Math 3B3 Winter 2004
Assignment 4
Questions 1 and 2 are statements in Euclidean geometry of circles and spheres.
1. Let i (for i = 1, 2) be a circle with center Oi and radius ri . Suppose 1 and 2 are disjoint
and that neither one is contained in the inter
Math 3B3 Winter 2004
Solutions to Assignment 3
Questions 4 and 5
Questions 4 and 5 relate to the Poincar disk model of Hyperbolic geometry.
e
4. In this question you will study Hyperbolic reections.
(a) Lemma. Let O be the center of a circle and let A, B
Math 3B3 Winter 2004
Assignment 3
Questions 1-3 are statements in Hyperbolic geometry and are not referring to any particular model
of this geometry. You need to refer to the Axioms of Hyperbolic geometry and their consequences
we derived in class when so
Math 3B3 Winter 2004
1
Solutions to Assignment 1
Note: Figures are in a separate le!
1. Suppose that we interpret the undened terms point and line as follows: points
are lines in R3 passing through the origin and lines are planes in R3 containing the
orig
Solutions to Assignment 1
1. Let D be a point between A and B. Show that for any point C not on the line AB,
either CD < AC or CD < BC.
Solution. We will prove this by contradiction. Suppose contrary to the statement
of the problem that CD AC and CD BC. T
Math 3B3 Winter 2004
Solutions to Assignment 3
Questions 1-3 are statements in Hyperbolic geometry and are not referring to any particular model
of this geometry. You need to refer to the Axioms of Hyperbolic geometry and their consequences
we derived in
Math 3B3 Winter 2004
1
Assignment 2
Questions 1 and 2 are problems in Neutral geometry. Make sure you justify each step in
your proofs by referring to the Axioms of Incidence, Betweenness or Congruence, or to the
Propositions of Neutral geometry we proved
Math 3B3 Winter 2004
1
Assignment 1
Problems 1-3 refer to Hilberts axioms. Make sure you justify each step in your proofs by
referring to the Axioms of Incidence, Betweenness or Congruence, or to the Propositions
we proved in class.
1. Suppose that we int
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MATH 3B03
Assignment 1
Due September 26, 2011
1. Find the Cartesian equations for the following parametrized curves.
(i) (t) = (1 + cos(t), sin(t)(1 + cos(t).
(ii) (t) = (t2 + t3 , t3 + t4 ).
2. Let : R R3 be the curve (t) = (cosh(t), sin(t), cos(t).
(i)