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Geometry and Topology

Unit square

Is it possible to partition a unit square [0, 1] X [0, 1] into two disjoint connected subsets A and B such that A and B contain opposing corners? I.e., such that A contains (0, 0) and (1, 1), and B contains (1, 0) and (0, 1)? *----0 | | | | 0----* Evidently, A and B couldn't be path-connected because a path running fr

Fixed Point Theorem and Closed Unit Ball in Euclidean Space

The Brouwer Fixed-Point Theorem Let denote the closed unit ball in Euclidean space : . Any continuous map from onto itself has at least one fixed point, i.e. a point such that . Proof Suppose has no fixed points, i.e. for . Define a map , , by letting be the point of intersection of and the ra

Lebesque Number and Connectivity

Lemma. Let {Ui} be an open covering of the space X having the following properties: (a) There exists a point x0 such that x0 Ui for all i. (b) Each Ui is simply connected. (c) If i≠j, then Ui Uj is arcwise connected. Then X is simply connected. Prove the lemma using the following approach: To pro

Proofs : Collinear and Distinct; Boomerang Quadrilateral

1- Prove that if AF/FB = AF'/F'B where A, B, F, F' are collinear and distinct then F does not have to equal F' 2- Suppose that the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by a line at the points A' B' C' D' respectively, show that AA'/A'B * BB'/B'C * CC'/C'D * DD'/D'A = +1

Parallelograms

(See attached file for full problem description) Please complete 17-24.

Writing Equations from Word Problems : Time and Distance, Two Moving Objects

A Car leaves Oak Corner at 11:33 a.m traveling south at 70km/h. at the same time, another car is 65 km west of Oak Corner traveling east at 90km/h. a. Express the distance d between the cars as a function of the time t after the first car left Oak Corner. b. show that the cars are closest to each other at noon.

Ratios and contribution analysis

Problem 1 Prepare financial analysis of Panorama. The analysis should include a summarization of the ratio analysis, explanations of what those ratios tell about the financial condition of Panorama, and a summarization of the financial strength and weaknesses of Panorama. What information do I need to use for the Financia

Shell method of finding volume of revolving solid

Revolving the solid about the y-axis, use the shell method to find the volume of the solid bounded by: the line x=√2/2 on the right y=1/√(1-x²) above. The shaded area in the drawing is bounded by the y-axis to the left, and the x-axis on the bottom.

Shell method to find volume about x-axis

Use the shell method formula to find the volume of the solid generated by revolving the shaded region bounded by the curves and lines below about the x-axis: V=∫2π(shell radius)(shell height)dy = ∫2πx f(y)dy a≤y≤b Shaded region:: x=y² x= -y y=2

Volume of a solid of revolution

Find the volume of the solid generated by revolving the region described about the Y axis: Between (0,0) and (0,2), the triangular region between those points on the y-axis and the straight line x=3y/2 using the formula V=∫π[R(y)]²dy

The surface area paint needed

--- The surface area A of a steel cylinder is given by the formula A= 2.pi.r²+2.pi. r h Where r is the radius and h the height a) find the required radius if A = 12500mm² , and h = 150mm b) Determine the number of 2.5 liter tins of paint that are needed to coat 500 cylinders with a thickne

Kilometers and starting point

Can you tell me what the answer would be if I traveled north for 35 kilometers then traveled east 65 kilometers. How far am I from my starting point? using: 35^2+65^2?

Relation in radius and height of a cylinder of given volume

Suppose the volume of a cylinder (think about the volume of a can) is given by V = πr2h where r is the radius of the cylinder and h is the height of the cylinder. Now suppose the volume of the can is 100 cubic centimeters. How do would you write h as a function of r? What is the measurement of the height if the radius o

Please explain in steps how to complete the following problem

Find sin2x, cos2x, and tan 2x under the given conditions. 23. sinx=5/13 (0<x<pie/2) ans. sin 2x = 120/169, cos 2x = 119/169, tan 2x = 120/119 Please explain in detail step by step how to come up with this ans. 25. cos x=-3/5 (pie<x< 3pie/2) ans. sine 2x=24/25, cos2x=-7/25, tan2x=-24/7

Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r

Diagonals, Squares Circles and Endpoints

6. If a square's diagonal has endpoints (3,4) and (7,8) , find the endpoints of the other diagonal. (please illlustrate) What is the length of the diagonal? (Please give formula) What is the perimeter of the square? Write the equation of the circle that has the endpoint (3,4) as its center and goes throught the two c

Can you lease check my work

I have attached a file regarding contrapositives and inverse and converse statements --- Statement: If two lines are parallel then they do not intersect. True Converse: If 2 lines do not intersect then they are parallel False Inverse: If 2 lines are not parallel then they intersect

Space of functions is sequentially compact

Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions

Trivial Topology, Continuity and Connectedness

Let X and Y be topological spaces, where the only open sets of Y are the empty set and Y itself, i.e., Y has the trivial topology. ? Show that any map X --> Y is continuous ? Show that Y is path connected and simply connected. ---

Time And Distances

The movement of two submarines are being followed by a tracking system, and the positions of the submarines are modelled by points. The position of Sub A at time t is (2t+2, 2t+1) and the position of Sub B at time t is (4-t, t+5) (distance in Kilometers) 1.How would I go about eliminating t from each pair of coordinates, and

Homeomorphism

(See attached file for full problem description with proper symbols) --- Let and a map, given by . Let ~ be the equivalence relation on Xx[0,1] defined by and all other points are equivalent only to themselves. Show that Xx[0,1]/~ is homeomorphic to the Moebius strip. ---

Continuous and identification maps

(See attached file for full problem description with proper symbols and equation) --- Let be a surjective continuous map between topological spaces. Show that: a) If f is an identification mp, then for any pace Z and any map the composition is continuous if and only if g is continuous. b) If, for any space Z and any

Identification map

(See attached file for full problem description with proper symbols and equations) --- ? Let be the subspace of of all positive real numbers. Show that the map defined by is an identification map. ---