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.
It is well known that:
The shortest distance from a point to a line is the length of the line segment which is perpendicular to the line and joins to the point.
Who first proved this? How far back in time does it go?
Homework Statement
How do I find the surface area of a sphere (r=15) with integrals.
Homework Equations
Surface area for cylinder and sphere A=4*pi*r2.
The Attempt at a Solution
I draw the graph for y=f(x)=√(152-x2). A circle for for positive y values which I rotate. I will create infinite...
Homework Statement
A short paper 12-16 pages I'm also fairly new to this topic
Homework EquationsThe Attempt at a Solution
I tried to ecplain it's application using the Robertson walker mrttuc but it ended upmlooking too much like a physics psper:/
I'm in 11th grade right now, and I would like to know whether or not I should spend my time learning geometry as my high-school education system places zero emphasis on geometry. If so, what type of geometry should I start with? (euclidean, analytic, differential, non euclidean?)
by the time...
Homework Statement
Problem statement uploaded as image.
Homework Equations
Arc-length function
The Attempt at a Solution
Tangent vector:
r=-sinh(t), cosh(t), 3
Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use...
I have a certain Ansatz for a gravitational wave perturbation of the metric h_{\mu \nu} that is nonzero near an axis of background flat Minkowski spacetime
The Ansatz has the following form:
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 &...
ive been reading about people thinking sub Planck scale spacetime is "non commutative". i have a vague idea of non commutative geometry mathematically, but there's no "space" in non commutative geometry so how can SPACEtime be a non commutative ring?
i thought non commutative geometry was just...
Hello, i am Ali doing PhD studies i am working on Activation Analysis of Hybrid reactor but i just stat studying this geometry but i am not able to understand this geometry for MCNPX modeling, so i need some help in modelling this geometry in MCNPX.
please guide me in this regard
I was lying awake the other night and thinking about Pi and flatlanders. I haven't done a lot of topology reading, so forgive my naivete.
Pi on a flat surface is a number we know well, but what happens to the ratio of a circle's diameter to its circumference on curved surfaces?
First question...
I'm seeing a presentation of Euclidean geometry that isn't hand-holdy. I've looked at some textbooks used in high schools these days, and it's hard to find the axioms and theorems in the midst of all the condescension. I just want something that states the definitions, axioms and basic...
Homework Statement
Requirements: http://i.imgur.com/2WKyhto.png
Homework Equations
2(L * W + L * H + W * H)
SA = (2 * pi * radius * height + 2 * pi * radius^2)
height = (volume)/(pi * radius^2)
The Attempt at a Solution
Code link: http://pastebin.com/sKFEGN0C
In the figure, ABCD is a square of side 1 cm. ABFE and CDFG are trapeziums(/trapeziods?). The Area of CDFG is twice the Area of ABFE. Let x cm be the length of AE.
(a) Express the length of FG in terms of x.
(b) Find the value of x, correct to 2 decimal places.
Thanks! :D
Hello
1. Homework Statement
We define the Dupin indicatrix to be the conic in TPM defined by the equation IIP(v)=1
If P is a hyperbolic point show:
a. That he Dupin indicatrix is a hyperbola
b/ That the asymptotes of the Dupin indicatrix are given by IIP(v)=1
, i.e., the set of asymptotic...
Hey JO,
You all know the binomic formulas I guess. Let's look at the first:
(a+b)^2=a^2+2ab+b^2
Now this can be interpretet as the area of a square with the sides (a+b). And that means the area of the square is decomposed into the components a^2,2ab and b^2. And this can also be done for a cube...
Crossing over the following paragraph:
There are three types of special manifolds which we shall discuss, related to the real scalars
of gauge multiplets in D = 5, the complex scalars of D = 4 gauge multiplets and the
quaternionic scalars of hypermultiplets. Since there are no scalars in the...
I am currently in year 9 (9 grade for those in US) and I have a really rusty and a weak math background. I have 2 months of summer holidays coming up. I should be done with pre algebra in mid December. During my summer holidays I have more than 50 hours a week avalible for study and I was just...
Hi,
This is also a sort of geometry question.
My textbook gives a proof of the relation: sin(θ + Φ) = cosθsinΦ + sinθcosΦ.
It uses a diagram to do so:
http://imgur.com/gLnE2Fn
sin (θ + Φ) = PQ/(OP)
= (PT + RS)/(OP)
= PT/(OP) + RS/(OP)
= PT/(PR) * PR/(OP) + RS/(OR) * OR/(OP)
= cosθsinΦ +...
Homework Statement
Consider a universe described by the Friedmann-Robertson-Walker metric which describes an open, closed, or
at universe, depending on the value of k:
$$ds^2=a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]$$
This problem will involve only the geometry of space at...
Urs Schreiber submitted a new PF Insights post
Higher Prequantum Geometry V: The Local Observables - Lie Theoretically
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Urs Schreiber submitted a new PF Insights post
Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively
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I had geometry quite a while ago and I wonder if anyone has any idea how to tackle this problem:
Is there any ABCDS pyramid (where ABCD is a rectangle) in which each 2 edges have different lengths and |AS|+|CS|=|BS|+|DS|
Thanks
Urs Schreiber submitted a new PF Insights post
Higher Prequantum Geometry III: The Global Action Functional - Cohomologically
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Urs Schreiber submitted a new PF Insights post
Higher Prequantum Geometry II: The Principle of Extremal Action - Comonadically
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Homework Statement
The moment of the couple is 600k (N-m). What is the angle A?
F = 100N located at (5,0)m and pointed in the positive x and positive y direction
-F = 100N located at (0,4)m and pointed in the negative x and negative y direction
Homework Equations
M = rxF
M = DThe Attempt at a...
Urs Schreiber submitted a new PF Insights post
Higher Prequantum Geometry I: The Need for Prequantum Geometry
Continue reading the Original PF Insights Post.
Dear all
I am studying general relativity and i have a question as follow. We have the 2- sphere can be scanned totally by a coordinate system (theta, phi) with the metric tensor written in terms of theta and phi. Now i want to divide the 2-sphere into charts 4 charts then each will have its own...
Hi,
1. Homework Statement
C : ℝ→ℝ3 given by
C(t)= ( 1/2 [ (1+k)/(1-k) cos((1-k)t) - (1-k)/(1+k) cos((1+k)t) ] ; 1/2 [ (1+k)/(1-k) sin((1-k)t) - (1-k)/(1+k) sin((1+k)t) ] )
with 0<|k|<1
Show that C(t) is an epitrocoid and find R, r and d according to k
Homework Equations
Parametrization of...
Hope I am on the right forum (and that my question makes some sense-so here goes.
Imagine we are a race of people living on a sphere (not hard because we are)
However , rather than buying into the idea that lines are ideally straight we are and have always been well aware of how "parallel...
Homework Statement
What is the area of a triangle on Earth that goes from the North Pole down to the equator, through the prime meridian, across the equator to 30 degrees east longitude, then back up to the equator? The radius of the Earth is about 6378 km.
Homework Equations
alpha + beta +...
Hello,
I am totally bad at geometry , by geometry I mean plane euclidean geometry with similarities and circles. I sometimes feel totally lost with problems. For example:
The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A...
Hello,
I started my class on geometry a couple weeks ago, and I feel that I'm struggling. I took Algebra 1 and 2 over the summer, and I never had a problem. I passed both of those expedited classes never getting less than an A on any assignment, but I got a C on my first geometry test. I'm not...
I'm studying General Relativity and Differential Geometry. In my textbook, the author has written ##x^2=d(x,.)## where d(x,y) is distance between two points ##x,y\in M##. I couldn't understand what d(x,.) means. Moreover, I am not sure if this is generally true to write ##x^2=g_{\mu\nu} x^\mu...
Homework Statement
The Tensions in four cables are equal:
|T1| = |T2| = |T3| = |T4| = T
Determine the value of T so that the four cables exert a totl force of 12,500-lb magnitude on the support.
(Inser a picture of 4 connected cables angled 9, 29, 40, 51 degrees respectively)
My question...
Homework Statement
Three charged particles are placed at each of three corners of an equilateral triangle whose sides are of length 2.7 cm . Two of the particles have a negative charge: q1 = -6.0 nC and q2 = -12.0 nC. The remaining particle has a positive charge, q3= 8.0 nC . What is the net...
I am looking to pick up one of these texts, but I don't really want to buy all three. Is there a considered favorite? Thanks in advance.
B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity
T. Frankel: The Geometry of Physics
B. Schutz: Geometrical Methods of Mathematical Physics
Homework Statement
Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
In Euclidean geometry, the Cauchy-Schwarz inequality is ##|e\cdot f| \le |e||f|##. In a Minkowskian signature, the inequality is reversed for timelike vectors. Apparently for spacelike vectors it depends on whether the two vectors span the light cone...
The following passage has been extracted from the John Stewart's English translated version of the "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms" http://archive.org/details/sirisaacNewtons00stewgoog:
Here Newton...
Homework Statement
As shown in the diagram below, the shape consists of a square and a circle with centre Q. Given that QM = 3 cm, prove that MN = 6 cm.
Known data:
-- triangles APB and BQC are congruent
-- angle BMC = 90
-- triangles BMQ and BNA are similar and right-angled
2. Homework...
I've got this project that I'm having trouble pinning down.
I'm cutting a pic up into squares to make a sort of jigsaw puzzle. I want the pieces to be a pleasing shape and the number of pieces to be within a small range.
- proportions of pic: 3:2 (so, say 6000 wide by 4000 high) this is...
Hello guys .
Through all the analysis of theory of general relativity we used what so called Manifolds
Manifolds as we know are topological spaces that resemble ( look like) euclidean space locally at tiny portion of space
And an euclidean space is the pair ( real coordinate space R^n , dot...
I'm in my last semester of my undergraduate majoring in mathematics (focusing on mathematical physics I guess - I'm one subject short of having a physics major) and am wondering, largely from a physics perspective if it would be better to do a functional analysis course or a differential...
I have finished reading both books on geometry by Kiselev and now look to move on but can't find any book to let me do so.
Specifically, I would like a nice comprehensive book that goes into detail on transformations, isometries, coordinate geometry, symmetry, mensuration, and vectors. Now I'll...
The problem
a) find the length of AD in the figure
b) Why are there two solutions in a) and what solution fits the figure?
Figure
The attempt
I started with drawing a "help line" in the figure.
The cosine formula for AC with respect to triangles ADC and CBA gives us two equations:
##...
What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too.
Elementary Algebra:
Gelfand's Algebra
Gelfand's Functions & Graphs
Burnside's Theory of Equations
Euler's Analysis of the...