Chapter 6 Alternative Concepts for Parallelism: NonEuclidean Geometry
6.1
6.2
6.3
6.3
Historical Background of Non-Euclidean Geometry
An Improbable Logical Case
Hyperbolic Geometry: Angle Sum Theorem
Hyperbolic Geometry Enrichment and Workbook
Homework 7
Strogatz 2.1.2
*
In the next three exercises, interpret x sin x as a flow on the line.
2.1.2 At which points x does the flow have greatest velocity to the right?
Graph of y sin x
The fixed points occur at x, x . Where x is even, we have stable fixed point
Strogatz 2.2.10
*
*
2.2.10 (Fixed points) For each of (a)( e), find an equation x fx with
the stated properties, or if there are no examples, explain why not. (In all
cases, assume that f x is a smooth function.)
a) Every real number is a fixed point.
At
Strogatz 2.2.1
*
*
Analyze the following equations graphically. In each case, sketch the vector field on
the
real line, find all the fixed points, classify their stability, and sketch the graph of x( t) for
different initial conditions. Then try for a few
Strogatz 2.4.4
*
2.4 Linear Stability Analysis Use linear stability analysis to classify the
fixed points of the following systems. If linear stability analysis fails
because f x 0, use a graphical argument to decide the stability.
2.4.4 x x 2 6 x
x x 2 6
Strogatz 2.4.2
*
2.4 Linear Stability Analysis Use linear stability analysis to classify the
fixed points of the following systems. If linear stability analysis fails
because f x 0, use a graphical argument to decide the stability.
2.4.2 x x1 x2 x
x1 x2 x
Strogatz 2.2.8
*
*
2.2.8 (Working backwards, from flows to equations) Given an equation ,
we know how to sketch the corresponding flow on the real line. Here you
are asked to solve the opposite problem: For the phase portrait shown in
Figure 1, find an eq
Strogatz 2.3.2
*
*
2.3.2 (Autocatalysis) Consider the model chemical reaction
A X k 1 2X
k 1
in which one molecule of X combines with one molecule of A to form two
molecules of X. This means that the chemical X stimulates its own
production, a process ca
Strogatz 2.2.7
*
Analyze the following equations graphically. In each case, sketch the
vector field on the real line, find all the fixed points, classify their stability,
and sketch the graph of xt for different initial conditions. Then try for a
few minu
Strogatz 2.1.3
*
*
2.1.3 (can use dfield to see more, use sinx for the RHS - use Matlab notation)
a dfield/pplane tutorial (hyperlink)
Stable fixed points are , 0, .
a) Find the flows acceleration x as a function of x.
First we must derive an equation con
Strogatz 2.4.9
*
2.4.9 (Critical slowing down) In statistical mechanics, the phenomenon
of critical slowing down is a signature of a second-order phase transition.
At the transition, the system relaxes to equilibrium much more slowly than
usual. Heres a m
Chapter 4 Euclidean Geometry
4.1 Euclidean Parallelism, Existence of Rectangles
4.2 Parallelism and Trapezoids: Parallel Projection
4.3 Similar Triangles, Pythagorean Theorem,
Trigonometry
4.5 The Circle Theorems
Homework 6
1
4.1 Euclidean Parallelism, Ex
Hint for the final.
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
15 points
15 points
10 points
15 points
15 points
30 points
Angle sum of triangles new proof
Summary of EG, HG, and SG essay
Matching 10 statements with 8 situations
A n
Homework assignment 3
3.1 3.2 8, 12 1, 2, 3, 4, 6, 9, 13 & 15
Enrichment: In Euclidean Geometry, circles have 3 possible relationships: they don't intersect at all they intersect in one point they can be internally tangent or externally tangent they inter
Homework 7
6.2
2, 8
Enrichment exercise: Do the Moment for Discovery in Absolute Geometry on page 433
Be sure to write it up nicely with illustrations and complete answers. Feel free to
check with any other student in the class to make sure youve got
it.
MATH 5397 HW4
Things that are the same in Euclidian geometry and Spherical Geometry
1.
Both systems are axiomatic in nature
2.
Both systems share the 16 neutral geometry axioms
3.
Both systems include polygons with 3 or more sides (Triangles, Quadrilatera
Math 5397 - 03
Summer 2010
Instructor:
Email:
Website:
Leigh Hollyer
[email protected]
www.math.uh.edu/~dog
Textbook:
College Geometry: a Discovery Approach, second edition
David C. Kay
ISBN 0-321-04624-2
Chatroom:
Testing:
link on my website
must be proctored.
2.1 An Introduction to Axiomatics and Proof
2.2 The Role of Examples and Models
2.3 Incidence Axioms for Geometry
Homework assignment 1
1
2.1
An Introduction to Axiomatics and Proof
An axiomatic system is a formalized construct that is used in business, r
3.1
3.2
3.3
Triangles, Congruence Relations, SAS Hypothesis
Taxicab Geometry: Geometry without SAS Congruence
SAS, ASA, SSS Congruence, and Perpendicular Bisectors
Absolute Geometry
We are building axioms that will result in one of two choices*: Euclidean
3.3 Spherical Geometry
Neutral Geometry (axioms below) is a structure that works for all of the Big Three
geometries: Euclidean, Spherical, and Hyperbolic. You just make certain choices
about the distance boundary and the situation about parallel lines an
1. Define corrosion
Corrosion is the degradation of a metallic element, it converts from a pure form into a nonmetallic compound.
2. Answer the following
a. Distinguish between oxidation and reduction
i. Oxidation: Refers to an element giving away electro