Math 454, Spring '08, homework assignment #1, due Jan 31 Problem 1 Let OA = 2a be the diameter of a circle (where O is the origin) and OY and AV be the tangents to the circle at 0 and A, respectively. A half-line l is drawn from O and meets the circl
5.4 solution.
c.
Let V(s) lim Vi(s). Then, by Part a, each Vi s) is perpendicular to the tangent plane at i(s) and,
d
thus, (since the tangent planes vary continuously) ds V s is perpendicular to the tangent plane Tp.
If W(s) is the parallel transport of
First, check directly that H
2
i by parallel transporting a vector tangent to the initial edge
of (as depicted in Figure 5.2 of the text, in the case is a triangle, and then extending to arbitrary
polygons as in Figure 5.5). Where at each vertex the amoun
p
Exterior of polygon
Figure 5.D. Large polygon on sphere.
Dissecting into triangles without adding any new vertices: On a sphere, if there are no convex vertices then the exterior of the polygon is convex. In this case, pick a point p that is the opposit
SOLUTIONS
PROBLEM 5.2. Dissection of Polygons into Triangles
In the literature there are many incorrect descriptions of dissections of polygons, several by wellknown mathematicians. For a discussion of these errors see the article: Chung-Wu Ho, Decomposit
PROBLEM 7.4. Exponential Map and Shortest Is Straight
c.
Proof of Problem 7.4.c:
1. Look at the path marked in Figure 7.2 (in the text) with parametrization
cos /2
sin
where r( ) = a cos and r
a cos /2 cos 2
r tan . Thus
y1 , r
r
)2
(r
h2
y2 , r
h2
( ) =
SOLUTION SUMMARY:
PROBLEM 7.3. Circles, Polar Coordinates, and Curvature
a.
Let f(r) h( ,r) |y1( ,r)|, then in order to determine the third Taylor approximation we need to
find f (0), f (0), f (0), and f (0). Since y( ,0) = p is constant, it follows that
SOLUTIONS
Chapter 3
Extrinsic Descriptions of Intrinsic Curvature
PROBLEM 3.1. Smooth Surfaces and Tangent Planes
*a.
If the surface is infinitesimally planar at p and the tangent planes vary continuously, then, for every
tolerance /4 there is a field of
Solutions: Chapter 2 Extrinsic Curves
193
If t(x) = f (p) + f (p)(x-p) is the equation of the line tangent to the curve (x,f(x) at the point
(p, f (p), then f (x) t(x) = cfw_ [f (x) f (p)]/(x p) f (p) (x p). By definition, if f is differentiable at p
then
192
Solutions: Chapter 2 Extrinsic Curves
As soon as (pn
q) is less than 0.5, then [pn
q]=0.
In this f.o.v. it is true that, whenever |xn y| < 1, then |[xn] [y]| 1. Thus, with the radius = 5000,
if we set the tolerance = 1/4999, then, as soon as |xn y| <
INTRODUCTION TO MANIFOLDS - V
Algebraic language in Geometry (continued).
Everywhere below F : M N is a smooth map, and F : C (M ) C (N ) the associated homomorphism of commutative algebras, F g = g F (F g)(x) = g(F (x). Let x M be a poin
INTRODUCTION TO MANIFOLDS - IV
Appendix: algebraic language in Geometry
1. Algebras. Definition. A (commutative associatetive) algebra (over reals) is a linear space A over R, endowed with two operations, + and , satisfying the natural axioms of a
Math 454, Spring '08, homework assignment #2, due Feb 14 Problem 1 The Peano curve is a continuous curve r(t) = (x(t), y(t), 0 t 1 which passes through every point of the unit square 0 x, y 1. Explain why it cannot be rectifiable. (Yes, such path
Math 454, Spring '08, homework assignment #3, due Feb 28 Problem 1 Let C be a regular parametrized simple space curve of class C 3 . Prove that for any point P on C, there exists a plane passing through P and such that all nearby points on C lie on
Math 454 7 February, 2008
Isoperimetric Inequality: 4A L2 . This is for a simple closed curve (positively oriented).
Jordan Curve Theorem: Such a curve (simple closed) divides the plane into two regions: abounded interior and unbounded exterior.
INTRODUCTION TO MANIFOLDS - I
Definitions and examples
1. Topologic spaces Definition. A topological space M is an abstract point set with explicit indication of which subsets of it are to be considered as open. Such open-by-definition subsets are
INTRODUCTION TO MANIFOLDS - I
Supplementary problems
Matrix manifolds Problem 1. that the map Let M = Matnn Rn be the set of all square matrices. Prove det C = 0,
2
AdC : A C -1 AC,
defines a diffeomorphism of the manifold M onto itself. Probl
INTRODUCTION TO MANIFOLDS - II
Tangent Bundles
1. Tangent vectors, tangent space. Let M be a smooth n-dimensional manifold, endowed with an atlas of charts x : U Rn , y : V Rn , . . . , where M = U V are domains of the corresponding charts.
INTRODUCTION TO MANIFOLDS - III
Algebra of vector fields. Lie derivative(s).
1. Notations. The space of all C -smooth vector fields on a manifold M is denoted by X(M ). If v X(M ) is a vector field, then v(x) Tx M Rn is its value at a point x M
FORMS AND INTEGRATION - I
Differential forms: definitions
Part I: Linear Theory Let V R be a linear space: we avoid the symbol Rn since the latter implicitly implies some coordinates. Definition. An exterior k-form on V is a map : V V R,
k ti
HOMOTOPY FORMULA. COHOMOLOGY.
Analysis versus topology
1. Homotopy formula. Definition. The Lie derivative of a differential form d (M ) of any degree d on a manifold M n along a vector field v is 1 Lv = lim (g t - ), t0 t where g t is the flo
SOLUTIONS
Chapter 2
Extrinsic Curves
PROBLEM 2.1. Give Examples of F.O.V.s
a.
The following are a few of the examples that have been brought up in a class:
1. Consider successively a globe of the world, a map of the whole USA, a map of New York State, a m