Geometric Definition and 813 Threads

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, etc.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

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  1. M

    What is the sum of the two separate series in the original infinite series?

    Homework Statement Determine whether the serie is convergent or divergent , if it is convergent find its sum. Ʃ∞n=1 (1 + 2n )/ 3n Homework Equations Ʃa(r)n-1 = a / (1-r) r < 1 is converging or if r > 1 diverging The Attempt at a Solution Well I can see its a geometric series...
  2. Government$

    Geometric and arithmetic series

    Homework Statement If a,b,c, are at the same time fifth, seventh and thirty seventh member of arithmetic and geometric progression then a^{b-c}b^{c-a}c^{a-b} is: The Attempt at a Solution I tried solving system of equations but i have four unknown. I was able to reduce it to on unknown...
  3. dextercioby

    Geometric Description of Free EM Field

    I'm not a geometer, so I beg for indulgence on the below: In a modern geometrical description of electromagnetism (either in flat or in curved space-time*), I see at least 3 (or 4) (fiber) bundles over the 4D space-time taken to be the base space: * 1 the cotangent bundle and the bundle of...
  4. N

    What is an Example of a Matrix with Geometric Multiplicity Greater Than 1?

    after finding out what geometric multiplicity was, I was surprised to notice that in every question I'd done it was always 1. So I'm trying to prove an example with g.m. > 1 to see why it works. I've found a matrix which definitely has an eigenvalue with g.m. = 2. I've checked everything with...
  5. D

    Geometric Multiplicity of Eigenvalues

    Could someone please explain to me (with an example if possible) what is the Geometric Multiplicity of Eigenvalues? I cannot understand it from what I have read on the web till now. Thanks in advance.
  6. B

    Why Is My Calculation of the Geometric Series Incorrect?

    \sum^{\infty}_{k=4} \frac{1}{5^{k}} ar^n=a/(1-r) Here is what I did: a=5 r=1/625 (1/5)/(1-1/625)=(1/5)/(624/625)=625/3120=125/624 The Answer is 1/500 Where am I going wrong?
  7. B

    Geometric sequence question in IB HL mathematics paper 1 november 2010

    Homework Statement The sum S n of the first n terms of a geometric sequence whose nth term is u n is given by 7n-an / 7n Where a > 0 Find an expression for un Find the first term and the common ratio of the sequence Consider the sum to infinity of the sequence Determine...
  8. barryj

    A puzzling math geometric sequence question.

    Homework Statement This problem is taken directly out of a textbook. "The first three terms of a geometric sequence are 1,2, and 4. Susanna said the 8th term of this sequence is 128. Paul said the 8th term is 29. Explain how the students found their answers. Why could these both be...
  9. E

    What Is the Theoretical Yield of Geometric Isomers in This Lab Experiment?

    Homework Statement Calculate the theoretical yield of the isomers from your data. Note: Find the limiting reagent. List of Masses I obtained during lab: Maleic Anhydride (C4H2O3): 5.05g, molar mass=98.1g/mol Impure Fumaric Acid: 2.28g Pure Fumaric Acid(trans-C4H4O4): 1.45g Pure Maleic...
  10. Darth Frodo

    Geometric Optics: Solving for Position & Nature of Image

    Homework Statement An object is placed 400 mm in front of a convex lens of focal length 80 mm. Find the position of the image formed. State the nature of this image. A second convex lens of magnifying power X8 is placed 125 mm behind the first convex lens. What is the focal length of...
  11. P

    Arithmetic and Geometric Series (tortoise and the hare)

    Homework Statement The question is attachedk Homework Equations Sn = n/2[2a+(n-1)d] Sn = (a x (1-r^n))/1-r The Attempt at a Solution I already found the general formulas: Tortoise: Sn = n/2(40) Hare: Sn = (1000 x [1-0.5^n])/0.5 And I know that there tortoise will finish the...
  12. E

    Using radians to discover the lengths of geometric shapes (circles)

    Homework Statement refer to question image Homework Equations refer to question image againThe Attempt at a Solution refer to working out image This is my brothers maths homework. He normally doesn't use online methods to request help and this is his first time.
  13. Fredrik

    Geometric insight about rotations needed

    I'm trying to find a way to simplify a complicated proof. The worst step of the proof involves a product of five 4×4 matrices. I'm hoping, perhaps naively, that if I could understand why the result of this operation is so simple, I may be able to explain the proof to others without actually...
  14. P

    MHB Geometric meaning of Noether normalization theorem

    Let $X\subset \mathbb{A}^n$ be an affine variety, let $I(X)=\{f\in k[X_1,\ldots,X_n]:f(P)=0,\ \forall P \in X\}$. We consider the ring $$A=k[a_1,\ldots,a_n]=\frac{k[X_1,\ldots,X_n]}{I(X)}$$ where $a_i=X_i \mod I(X)$.Noether normalization says that there are algebraically indipendent linear forms...
  15. B

    How is the Chain Rule Applied in Geometric Tangent Vectors?

    So let ℝ^{n}_{a}={(a,v) : a \in ℝ^{n}, v \in ℝ^{n}} so any geometric tangent vector, which is an element of ℝ^{n}_{a} yields a map Dv|af = Dvf(a) = \frac{d}{dt}|_{t=0}f(a+tv) this operation is linear over ℝ and satisfies the product rule Dv|a(fg) = f(a)Dvg + g(a)Dvf if v|a =...
  16. K

    Geometric series with modified terms

    I am looking for a way to sum some numbers. I understand that if I want to sum pi, I can use the geometric series: \sum\limits_{i=0}^N p^{i} = \frac{1-p^{N+1}}{1-p} But can anyone help me with what to do when I need: \sum\limits_{i=0}^N p^{i} q^{ti} where t is just a constant...
  17. MarkFL

    MHB Aesity's question at Yahoo Answers regarding a geometric sequence

    Here is the question: Here is a link to the question: Geometric sequence question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  18. A

    MHB Formula for a geometric series with variable common ratio

    I have the following summation where a is a positive constant and can be > 1 $$ \sum_{k=2}^{n}a^{n-k} $$ I am trying to find a general formula for this summation, which turns out to be a geometric series with a as the common ratio. I have worked out the following: $$ \sum_{k=2}^{n}a^{n-k} =...
  19. M

    Is AdS/CFT geometric or quantum?

    Hi everyone, long time lurker first time poster. There's been a some debate here on AdS/CFT and I can't resolve the facts of the matter. I have read Maldacena's large-N paper but of course there is no mention of AdS/CFT in there. The reason I'm posting is because I was hoping someone could...
  20. S

    Physics Calculations That Apply to a Variety of Geometric Objects

    I had a serious problem when I read my first book on calculus-based physics when I came to the center of mass & moment of inertia sections of my books. I really freaked myself out over the fact that even restricting ourselves to the most common & ideal geometric objects the list of things you'd...
  21. N

    Inequality of arithmetic and geometric means

    Hey! I have this: 2(√(1-a^2 ))+ 2a How to determine the maximum value of this? I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet. So, have you got any ideas? Poor Czech Numeriprimi... If you...
  22. N

    MHB Geometric Progression sequence with an Arithmetic Progression grouping problem

    Good Day, My friends and I are stuck on solving the last part of the attached problem. The solution is 2^[(n^2 + n)/2] - 1. Can anyone help us with solving this? Thanks & Regards, Nicodemus
  23. Lebombo

    What is the geometric property of these 2 angles being equal?

    At time 2:20, the woman says, "Using trigonometry, we know that this angle is the same as this angle." Which trigonometry or geometry property is she referring to that is needed in order to determine those two angles are equal? Is there a name for it? I don't have a geometry book, but I am...
  24. shounakbhatta

    Orthogonal set - Geometric interpretation

    Orthogonal set -- Geometric interpretation Hello, If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0. Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero. Secondly, if I have three vectors...
  25. H

    Solving for Parameter Restrictions in Q2: Geometric Help Needed

    Homework Statement The problem is Q2 in the attached Homework Equations The Attempt at a Solution I am trying to determine the rigion of parameters for each number of crosses,where x and y are the distances of the center of cross from the closest line respectively,and θ is the acute angle.I...
  26. S

    Help Geometric Objects Thin lenses Problem

    I really need help with this problem. Give me at least a hint on how to solve it. You are standing in front of a lens that projects an image of you onto a wall 2.00m on the other side of the lens. This image is three times your height. A) How far are you from the lens? B) What is...
  27. Y

    Sum of N geometric variables with changing probability

    Homework Statement Ʃ(A-i)/(N+1-i) sum of i=1 to N Homework Equations How do I solve this series for all 0<N<A cases. This series is the sum of N geometric variables of changing probability. I'd appreciate any help
  28. G

    Torque- Vector cross product using both geometric and algebraic methods

    Homework Statement A lever is orientated along the y direction in a Cartesian coordinate system. The length of the lever is 0.5m and one end of it is at the origin of the coordinate system. A (3i-5j)N force applied to the other end of the lever. Calculate the Torque produced by the force...
  29. L

    Comparison Test and Geometric Series

    The Problem: Let a(n) = (n^2)/(2^n) Prove that if n>=3, then: (a(n+1))/(a(n)) <= 8/9 By using this inequality for n = 3,4,5,..., prove that: a(n+3) <= ((8/9)^n)(a(3)) Using the comparison test and results concerning the convergence of the geometric series, show that: The...
  30. D

    MHB Problem involving arithmetic and geometric mean.

    $a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that $$ab^2c^3 \leq \frac{1}{432}$$
  31. V

    Infinite Geometric Sequence: How to Find the Number of Terms

    Homework Statement How many terms are in each sequence? 12, 4, 4/3, ..., 4/729 Homework Equations The Attempt at a Solution using tn=t1(r)(n-1) ? I am lost
  32. T

    Really silly Geometric Progression question

    Homework Statement I should know this (it's been a few years) but can't seem to get the answer for: Ʃ (from i = 0 to n) 0.5-i the answer is apparently 2n+1-1 / 2 - 1 how do I go about getting this? Homework Equations I tried Ʃn-1k=0ark = a* 1-rn / 1 - r with no luck The Attempt at a...
  33. Femme_physics

    Geometric Tolerances: Standards & Definitions

    Geometric Tolerances - Do standards for defining "general geometric tolerances" exist For example, if I want to define non-geometric tolerances for the whole part, I just write what type of IT it is. For instance, IT8. And then the manufacturer just looks at the chart to know the tolerances he...
  34. H

    Proof of 2nd Derivative of a Sum of a Geometric Series

    Homework Statement I am trying to prove how \(g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}\). I don't know what I am doing wrong and am at my wits end. The Attempt at a Solution (The index of the summation is always k=2 to infinity)...
  35. shounakbhatta

    Determinant and geometric representation

    Hello, Typically the area of a parallelogram if give by A=b*h The det(M) =ad-bc, where m=2x2 matrix. How they are related? -- Shounak
  36. D

    Using two sums to find geometric sequence

    Hi. I'm currently tutoring this student with High school math, and I'm completely stumped on this question that he was asked on his test. I'm hoping the community can help me help my student! Homework Statement The student was presented with two sums of a geometric sequence (eg, Sum of...
  37. H

    Geometric Distribution Probability problem

    Homework Statement We roll a fair die until we get a three or a four. Z denotes the number of rolls needed. What is the probability that Z >= 3? (replacement assumed) Homework Equations Geometric distribution seems logical here? The Attempt at a Solution Let p(A) = p(getting a...
  38. mnb96

    Geometric interpretation of metric tensor

    Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.
  39. N

    Geometric density Statistics/Probability

    Homework Statement Suppose that X_1, X_2, ... are identical and independently distributed with F_x1(x) = exp(x)/(1+exp(x)) for -infinity < x < infinity Suppose that N independent of X_i has geometric density f_N (n) = P(N=n) = p(1-p)^(n-1) for n =1,2,3,... and 0<p<1 Let Z = max{X_1...
  40. P

    Geometric Optics - Converging Lens

    Homework Statement A certain lens focuses light from an object 2.90m away as an image 46.9cm on the other side of the lens. 1. What type of lens is it? (Converging or Diverging?) 2. What is the focal length? 3. Is the image real or virtual? Homework Equations 1/di + 1/do = 1/f...
  41. B

    Geometric Series Test Rather Than Integral Test

    Homework Statement \sum_{n=1}^{\infty} \frac{1}{2^n} Homework Equations The Attempt at a Solution Could I some how manipulate this to fit a geometric series, so that I may instead use the geometric series test?
  42. D

    MHB Are Geometric Progressions Always Less Than or Equal to Arithmetic Progressions?

    Is it true that geometric progressions are \leq arithmetic?
  43. I

    Sum of this geometric sequence doesn't make sense

    Homework Statement 14 Ʃ 2(4/3)^n n=1 Homework Equations Sn=a(1-r^n)/(1-r) The Attempt at a Solution 2(1-[4^14]/[3^14])/(-1/3)=330.74 However, the answer sheet gives ~441 as the answer, and I confirmed it by doing it by hand. Why is the equation not working? What's wrong?
  44. D

    Geometric Algebra, prerequisites to studying? Anyone here study it?

    I was curious if anyone here ever studied Geometric Algebra? It seems not so mainstream and fairly new and I feel intrigued by the subject but I don't want to get in over my head. Just browsing through the table of contents of some books has a lot of unfamiliar terms to me. Here was a...
  45. sankalpmittal

    Questions of geometric progression.

    Homework Statement 1. Prove that 1111111...65 times is not a prime number. 2. If a,b,c,d are in GP , then (a2+b2+c2)(b2+c2+d2) is : (a) (ab+ac+bc)2 (b) (ac+cd+ad)2 (c) (ab+bc+cd)2 (d) None of the above Homework Equations Formulas of GP : 1.Sn = a(1-rn)/(1-r) 2.a,b,c in GP...
  46. D

    Dot Product/Cross Product Interpretation, Geometric Construction

    Homework Statement Given the nonzero vector a ε ℝ3, a\dot{}x = b ε ℝ, and a × x = c ε ℝ3, can you determine the vector x ε ℝ3? If so, give a geometric construction for x. Homework Equations a\dot{}x = ||a||||x||cos\Theta The Attempt at a Solution I'm not really certain what it is...
  47. B

    Simplifying finite geometric series expression

    Homework Statement I've come across the type of sum in several places/problems but seem to be making no progress in trying to simplifying it further. We have a finite series of some exponential function. \sum_{n=0}^{N}e^{-na} Where a is some constant, a quantum of energy or a...
  48. S

    Is the Series Ʃ (1 + 2^n) / 3^n Convergent and What is Its Sum?

    Homework Statement Determine whether the series is convergent or divergent. Find the sum if possible Ʃ 1+2^n / 3^n n=1 -> infinity Homework Equations a/1-r The Attempt at a Solution I split it up so that the equation is now: Ʃ (1/3^n) + (2/3)^n n=1 -> infinity Ʃ (1/3^n)...
  49. P

    Simple Geometric Proof: Inscribed Triangle in a Circle of Diameter d

    Consider a circle of diameter d. Inscribe a triangle within the circle so that the triangle has hypotenuse d. Prove that this triangle is always right angled. If we define the 3 points on the circumference of the circle that define the triangle as A B and C such that |AC| = d then we...
  50. A

    Minimum distance between a point and a geometric locus

    Hi guys I have a problem to solve, I'd like to find the minimum distance between a point and a geometric locus described in closed form, for example the intersection of two circles: p= point coordinate p1= center coordinate circle 1 p2=center coordinate circle 2 r1=radius of circle 1...
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