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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

Synthetic Geometry

Postings: 55

Synthetic geometry differs from other branches of geometry because it focuses on pure geometrical contents and draws conclusions through the use of axioms, logical arguments and theorems.

Algebraic Geometry

Postings: 98

Algebraic geometry is a field of mathematics which combines two different branches of study, specifically algebra and linear algebra.

Analytic Geometry

Postings: 1,265

Analytic geometry is a field of geometry which is represented through the use of coordinates which illustrate the relatedness between an algebraic equation and a geometric structure.

Geometric Shapes

Postings: 569

Geometric shapes are figures which can be described using mathematical data, such as equations, and are an important component to the study of geometry.

Differential Geometry

Postings: 11

Differential geometry is a field of mathematics which possesses similarities to the study of calculus, but differs in how it applies the techniques of integration and differentiation to more complex, higher dimension problems.

Manifolds

Postings: 11

A Manifold is a topological space which is locally Euclidean; meaning that the vicinity around each point resembles Euclidean space.

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<=

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

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

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

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

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

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

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

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

Metric Tensors and Christoffel Symbols

Problem 1. Derive the formula given below for the Christoffel symbols ?_ij^k of a Levi-Civita connection in terms of partial derivatives of the associated metric tensor g_ij. ?_ij^k = (1/2) g^kl {?_i g_lj ? ?_l g_ij + ?_j g_il }. Problem 2. Compute the Christoffel symbols of the Levi-Civita connection associated to ea

Find the Area of a Rectangle

For your assignment this week, imagine that you will be building a shed in your back yard. The shed requires a cement foundation that is rectangular in shape. You would like to mark the location of the cement foundation to ensure that it is the correct size and shape. You do not have any special equipment that will help ensure t

Math for Water Distribution Operator Certificate

See the attached file. I'm taking a Water Supply Technology math class to get a Water Distribution Operator Certificate. We are covering Volume of Rectangular and Cylindrical Tanks, Pipelines, and Rectangular Channels. We have not covered things like flow rate as it relates to time as in detention time. We are not there yet

Quantitative Analysis of The Italian General's Pizza Parlour

1. You must show all steps including formulas used and all calculations done to arrive at the final answers. Incomplete solutions will receive partial credit. 2. Use at least four significant digits at all intermediate steps. Round off the final answers appropriately. Note: 0.0042 is only two significant digits as leading zer

Inventory Management

Shoe Shine is a local retail shoe store located on the north side of Centerville. Annual demand for a popular sandal is 500 pairs, and John Dirk, the owner of Shoe Shine, has been in the habit of ordering 100 pairs at a time. John estimates that the ordering cost is $10 per order. The cost of the sandal is $5 per pair. For Jo

Finding a Surface Area with Given Length, Width and Height

The following has me really stumped. Can You please help me with this problem?: Find the surface area of the following room measurements: LENGTH:8 feet *10 inches = 106 inches WIDTH: 12 feet * 9 inches = 153 inches HEIGHT: 7 feet * 10 inches = 94 inches Then: A gallon of paint covers about 350 square feet. How many g

volume of the solid generated by revolving

Please see attachment. Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis. y = x^2, y = 7-6x, x = 0, for x >=0

Word Problem

It is a common experience to hear the sound of a low flying airplane, and look at the wrong place in the sky to see the plane. Suppose that a plane is traveling directly toward you at a speed of 200 mph and an altitude of 3,000 feet, and you hear the sound at what seems to be an angle of inclination of 20 degrees. At what ang

Topology - open or closed susbset

Please help with the following problem. For the following, I'm trying to decide (with proof) if A is a closed subset of Y with respect to the topology, T (i) Y = N, T is the finite complement topology, A = {n e N | n^2 - 2011n+1 < 0}. (ii) Y = R, T is the usual topology, A is the set of irrational numbers between 0 and

Problems exemplifying congruency

1) Suppose n belongs to Z. (a) Prove that if n is congruent to 2 (mod 4), then n is not a difference of two squares. (b) Prove that if n is not congruent to 2 (mod 4), then n is a difference of two squares. 3) Let n = 3^(t-1). Show that 2^n is congruent to -1 (mod 3^t). 5) Let p be an odd prime, and n = 2p. Show that a^(

De Morgan's Laws - FIP, Hausdorff and Compact

1.Let X be a set and T and T' are two topologies on X. Prove that if T subset of T' and (X,T') is compact, then (X,T) is compact. Prove that if (X,T) is Hausdorff and (X,T') is compact with T subset of T', then T=T'. 2.Let X be a topological space. A family {F_a} with a in I of subsets of X is said to have the finite

Using Differentials to Estimate

1. In the manufacturing process, ball bearings must be made with radius of 0.5mm, with a maximum error in the radius of +/- 0.016 mm. Estimate the maximum error in the volume of the ball bearing. Solution: The formula for the volumen of the sphere is _____. If an error deltaR is made in measuring the radius of the sphere