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# Geometry and Topology

Geometry and topology are two distinct topics, in which the branch of geometry analyzes metric space and the study of topology investigates topological space. In Euclidean geometry, a set of elements existing within three dimensions has a metric space which is defined as the distance between two elements in the set. Conversely, topological space is a concept which considers Euclidean geometry and looks to generalize the structure of sets.

More specifically, in mathematics topological space is defined as a pair (X, T). In this pairing, X represents a set and T is a topology of a collection of subsets on X.  This set also has a set of particular properties such as T needing to encompass both X and the empty set. It is critical to understand the definition of a topological space so that proofs can be completed to identify different topologies, such as discrete and indiscrete topologies.

Simply put topology aims to elucidate upon the qualitative elements of geometrical shapes and structures. Geometry analyzes shapes and structures in flat space, such as circles and polygons and investigates the properties of these structures.  Whereas geometry is concerned with whether certain shapes may be congruent or not, topology considers different problems, such as whether these shapes are connected or separated.

Geometry and topology are two very important subjects in the discipline of mathematics. Furthermore, these topics extend into other mathematical areas such as combinatorics and algebraic geometry.

## Categories within Geometry and Topology

### Metric Space Distances and Radiuses

Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

### Global maximum of given set and extreme values

Please help with the following mathematics problems. (a) Let f be a differentiable functions defined on an open set U. Suppose that P is a point in U that f(P) is a maximum, i.e. f(P) >= f(X) for all X E U Show that grad f(P) =0 (b) Find the global maximum of the function f(x,y)=x^3 +xy defined on the set S={(x,y)|-1<=

### Invertible Functions and Local Inverses

See the attached function and answer the following questions: (a) Is f invertible? (b) In which points does f have a local inverse? (c) Determine the derivative of the local inverse at f((-1,1)) and f((1,-1))

### An Example Using the Chain Rule

Let f: R^3 --> R^2 by given by f((x,y,z)) = (x^2 + y^2*z^2 + y*z^2) and let g: R^2 --> R^2 be given by g((u,v)) = (log(1 + u^2v^2), u^3v) Use the Chain Rule to compute (g o f)' at (1,0,1)

### Global Maximum and Global Minimum

Consider the following function: f(x,y) = xy on the set S = {x^2 +4y^2 ≤ 1}. a) Explain by applying a relevant theorem why f(x,y) has a global maximum and a global minimum in the set S. b) Find the critical of f in the interior of the set S. c) Use the method of Lagrange multipliers to find the minima and maxim

### The Taylor Expansion of Multivariate Function

Given two position vectors x and h, Taylor's formula up to order two can be written as: (See attachment) a) Write down Taylor's formula in two variables (x,y) with h = (h,k)^T, using Di to denote partial derivatives. b) State the conditions that partial derivatives commute, namely, D1D2f = D2D1f (See attachment) c) Give

### Monthly and Annual Budgets

Steve is buying a farm and needs to determine the height of a silo on the farm. Steve, who is 6' tall, notices that when his shadow is 9' long, the shadow is 105' long. How tall is the silhouette?

### Surface and Partial Derivatives

Consider the function f (x, y) given by f (x, y) = ( 1 - x^2 - y^2)^2. a) Sketch the graph of the curve f (x, 0) for x e [-2,2]. b) Sketch the level curves for f (x, y) = c for c = 0, c = 1/4, c = 3/4 and c = 2. Also plot the level set for f (x,y) = 1. See attachment for full question.

### Parameterization of Curves

See attachment for equation and full problem set. a) Find a parametric representation of the tangent line to the curve C2 at t = 1 b) Derive the equation of the normal plane to the curve C2 at t = 1, of this form ax + by +cz = d c) Find the distance of the plan in (b) and the origin (0,0,0) d) Find the speed of the c

### Trigonometry of a Kite

Lines with special relationships to the sides and angles of a triangle determine proportional segments. When you know the length of some segments, you can use a proportion to find an unknown length. You are making the kite shown on the right (see attachment) from five pairs of congruent panels. In parts (a) - (d) below, use

### Centroids of triangles

The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + ve

### Theory of Probability and Statistics

Describe the historical development of statistics and probability. Please include some reference to non-Western mathematics history.

### Synthetic and analytical proof: parrallelogram

Theorem: If the consecutive midpoints of the sides of a parallelogram are joined in order, then the quadrilateral formed from the midpoints is a parallelogram. A. Prove the theorem given above in Euclidean geometry using synthetic techniques. 1. Include each step of your proof. 2. Provide written justification for each

### Euclid's parallel postulate, hyperbolic and spherical geometry

A. Discuss differences between neutral geometry and Euclidean geometry. B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.

### Mathematics

Task: A. Discuss differences between neutral geometry and Euclidean geometry. B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries. Note: Euclid's parallel postulate states the following: "For every line l and for every external

### Orthogonal Projection Operators

Let P1 be the orthogonal projector onto the subspace \$1, P2 the orthogonal projector onto the subspace\$2. Show that, for the product P1P2 to be an orthogonal projector as well, it is necessary and sufficient that P1 and P2 commute. In this case, what is the subspace onto which P1P2 projects?

### Dynamic Geometry

Given: It is important to know how to use dynamic, interactive software programs such as The Geometer's Sketchpad, Cabri Geometry, GeoGebra, or Google SketchUp, to improve the teaching and learning of geometry. Task: 1. Distinguish between static and dynamic geometry problems. 2. Use dynamic geometry software to c

### Geometric proofs using right angles

Need to write a proof for the following: Given: BAC is a right angle DEC is a right angle line DB bisects line AE Prove: C is the midpoint of line DB. I've attached the question that needs to be proved. I need the proof on the attached document called "proof(2). See attached PDF fi

### The solution gives detailed steps on calculating the reliability at given time. All formula and calcuations are shown and explained.

What is reliability at t=1? Reliability distribution where the failure rate is given by r(t)=0, t<0 , and r(t)= sin(t)+1, for t>0. What is reliability at t=1? See the attached Word document for the full problem, properly formatted.

### Geometry: Battle with the Squirrels

In my latest battle with squirrels, I have strategically hung my bird feeder so that a squirrel cannot steal my birdseed. I attach string to a branch 15 feet off the ground and 3 feet from the trunk. If I attach the other end to a circular spool of radius 1 foot that 3 feet off of the ground and 10 feet away from the base of the

### Logical progression of steps to develop a reasonable and complete geometric proof showing that S is the midpoint of line segment RT.

Please complete the appropiate proof using a 2-column proof format with statements on the left and reasons on the right. ** Please see the attached PDF document for the complete problem explanation and the associated diagram **

### Review on the Basic Topology Spaces

1. a) Suppose T_1 is a topology on X = {a,b,c} containing {a}, {b} but not {c}. Write down all the subsets of X which you know are definitely in T_1. Be careful not to name subsets which may or may not be in T_1. b) Suppose T_2 is a topology on Y = {a,b,c,d,e} containing {a,b}, {b,c}, {c,d} and {d,e}. Write down all the subse

### Topological Surfaces

1) The definitions of surface (in terms of gluing panels) and what it means for two surfaces to be topologically equivalent. 2) A description of the three features of surfaces that characterize them in terms of their topology. 3) Three examples of pairs of surfaces that agree on two of the features but differ on the third

### Music Spiral Formula for Fret-to-Fret Spacing

Hello. My name is Clark. I build musical instruments as a hobby and am building a stringed instrument that requires a spiral shaped gear. To generate this gear I need the geometry for the spiral (I can add the teeth). I have attempted to express the problem in the simplest way that I can. I have also attached a spreadsh

### Topology of Surfaces: Point-Set Topology in R^n.

I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below. PROBLEM (Exercise 2.26). Describe what stereographic projection does to (1) the equator, (2) a longitudinal line through the north and south poles, (3) a tr

### Problems in the Point-Set Topology of R^n

Let Fr(A) denote the frontier set of A and Cl(A) denote the closure of A, where A is a subset of R^n. Solve the following problems. Exercise 2.6: For any set A, Fr(A) is closed. Exercise 2.12: If A and B are any sets, prove that Cl(A and B) belongs to Cl(A) and Cl(B). Give an example where Cl(A and B) is empty, but Cl(A) a

### Co-Finite Topology

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X? 3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

### Fixed set under continuous map on a compact Hausdorff space.

The question we want to answer is as follows. For a nonempty compact Hausdorff topological space X and a continuous function f:X-->X we want to show that there is a fixed set A for f, that is, A is nonempty and f(A)=A. We also construct an example of a Hausdorff space X which is not compact for which there are no fixed sets,

### Number of connected components and continuous maps, Topology

The following question is answered: Let f:X-->Y be a continuous onto map. Let D be a subset of Y such that YD has at least n connected components. Prove that Xf^(-1)(D) has at least n connected components. Are the line and the plane with their usual topology homeomorphic?

### City Designer Project

City Designer Project Your city must have at least six parallel streets, five pairs of streets that meet at right angles and at least three transversals. All parallel and perpendicular streets should be constructed with a straight edge and a compass. Use a protractor to construct the transversal street. Name each street i