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.
Homework Statement
If u is a unit vector, find u.v and u.wHomework Equations
I assumed that unit vector means u=<1,1,1>
u.v=|u||v|cos60
My knowledge of unit vectors is very limited. I know that a unit vector is
i=<1,0,0>
j=<0,1,0>
k=<0,0,1>
The Attempt at a Solution
Since the triangle is...
I'm currently studying abstract algebra from Herstein's interesting book "Topics in algebra", I've learned different definitions so far and I've solved most of the problems covered in the book. I've so far studied groups, subgroups of them, normal subgroups, quotient groups, isomorphism...
Homework Statement
Ok I'm able to track one star in the sky, over a period of one hour. We use three measurements to find the angle of the arc traced by the star. The three measurements also constitute two vectors. We can take the cross product of those vectors and it will give us a vector...
I'm working with 3D geometry, and I've been at this for days. I'm beating my head against a wall, because I'm nearly done with the project. There's only one glitch in my system.
The Situation:
I have a 3D cartesian coordinate system with a Spherical system overlaid over it (the "poles"...
Homework Statement
a. Suppose you do not know your location (on Earth) or the direction of north. Now suppose you track one particular star in the sky. You measure its exact position in the sky and record the exact time of the measurement. How many such measurements are necessary to deduce...
I have decided to attempt to pick up some differential geometry on my own, and I am trying to get some traction on the subject which I do by trying to reduce it to familiar and simple cases.
This is a trivial case, I know, but it will go a long way in advancing my understanding. Suppose the...
Homework Statement
then points are uniformly spaced on a circle. Each of the points is connected by a segment to exactly one of the other points for a total of five segments. Some pairs of the segments may intersect and some may not. What is the maximum possible number of distinct intersection...
Hey Everyone,
I just wanted to ask for a bit of help on this research assignment I have to do. I have to show how Fractal Geometry contributes to the theory that Mathematics was invented. I have been looking into fractal dimensions and the fact that the dimensions we have labelled (1,2 and 3)...
Homework Statement
Homework Equations
L[c]:=\int_{a}^{b}(\sum_{i,j=1}^{2}g_{ij}(c(t))c_{i}'(t)c_{j}'(t))^{1/2}dt
The Attempt at a Solution
So g_{ij}(x,y)=0 for i{\neq}j, c_{1}'(t)=-Rsin(t), c_{2}'(t)=Rcos(t)
so...
The stress-energy tensor is associated to a volume density and flux in 4-spacetime and the Einstein tensor seems to represent a three-dimensional curvature (being a one-form with vector values) that acts on and is acted by the stress-energy source.
If this is correct, I'm not sure what is this...
String field from Loop SpinfoamQG:how to build SM matter on discrete quantum geometry
note: EPRL is the current standard spinfoam formulation of Loop Quantum Gravity.
http://arxiv.org/abs/1201.0525
String Field Theory from Quantum Gravity
Louis Crane
(Submitted on 2 Jan 2012)
Recent work...
Homework Statement
Given that P,Q and R lie on the hyperbola xy=c2, prove that if PQ and PR inclined equally to the coordinate axes, then QR passes through O.
Homework Equations
The Attempt at a Solution
I don't understand what does ''PQ and PR are inclined equally to the...
Homework Statement
(E) is a group of points M from a level/plane
MA^{2}-MB^{2}=-4 And I is the center of [AB]
Homework Equations
show that IM*AB=-2 ( IM and AB have arrows on top)
The Attempt at a Solution
Well i split MA^{2}-MB^{2}=(MA-MB)(MA+MB)
then i got ...
Homework Statement
2 circles have the equation x2+y2-2x-2y+1=0 and x2+y2-12x-12y+36=0 respectively. Both circle touches the x-axis, y-axis and the line 3x + 4y = 12. Find the fourth tangent of the 2 circles.
Homework Equations
The Attempt at a Solution
This is second part of the...
Homework Statement
A variable point P lies on the curve y2 = x3 and is joined to a fixed point A with coordinate (2,0). Prove that the locus of the mid-point of AP is y2= 2(x-1)3.
Homework Equations
The Attempt at a Solution
According to what i know, I need to know the...
Astronomers say that there is neither an edge or center to the universe, yet we live in a finite universe. I understand how our universe can be finite yet have no edges due to the curvature of the universe, but I can't understand how it doesn't have a center. Space must be being curved around...
Homework Statement
Disparity is defined as \delta = \alpha - \beta. Find \delta in terms of interocular distance a, viewing distance D and d, e and f.
http://img220.imageshack.us/img220/7576/43519392.jpg
The Attempt at a Solution
I'm not getting anywhere. Any tips to get me started?
So I took an analysis class which covered chapters 9 and 10 of Rudin's PMA, for those of you who don't know that's multivariable analysis and differential forms, and I have taken a course in vector calculus but never a proper course on differential geometry. Thus my introduction to the subject...
Hi, given the algebraic function:
f(z,w)=a_n(z)w^n+a_{n-1}(z)w^{n-1}+\cdots+a_0(z)=0
how can I determine the geometry of it's underlying Riemann surfaces? For example, here's a contrived example:
f(z,w)=(w-1)(w-2)^2(w-3)^3-z=0
That one has a single sheet manifold, a double-sheet...
Homework Statement
There are three problems based on the following diagram.
You have a triangle on the cartesian plane with each corner having the coordinates (-a, 0) (a,0) (b,c)
Find the coordinates (in terms of a,b,c) of the point where all three perpendicular bisectors of the...
This is a Lagrangian problem, I am posting it here in introductory physics because what I need help with isn't in Lagrangian mechanics, but rather geometry.
http://img97.imageshack.us/img97/7504/what3.png
I am confused as how they got those relations for x and y. I have tried to make...
Suppose you have the simplest type of black hole - time independent, no angular momentum or charge, Schwarzschild solution, then you modify the situation by adding a stationary mass outside the event horizon (I imagined lowering this slowly on an idealised string).
The question is, does this...
Hi all , How can I find lecture notes on ArXiv ? I was looking for lecture notes on Yang-mills theories treated in the language of differential geometry but didn't succeed till now . Can some one recommend me some good resource for it?
[/itex][/itex]Homework Statement
A battery consists of a cube of side L filled with fluid of conductivity s. The electrodes in the battery consist of two plates on
the base at y = 0, one grounded and one at potential V = 12 Volts. The other sides of the battery casing are not
conductive...
Homework Statement
I have this arrow head geometry question (Please see diagram). I know L1, L2 and L3, angle B and the constant k (notice how the corner angle is equal to k * t2). I don't know t1 and t2. I'm pretty sure I have enough constraints, I'm just having trouble finding the right...
Homework Statement
Please see below...
Homework Equations
Please see below...
The Attempt at a Solution
Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution...
Thank you...
Homework Statement
I have a three link revolute manipulator at the origin. I know all the link lengths. The joint angle for the third link is coupled to the second such that Theta3 = k*Theta2. I want to determine the joint angles (thetas) of the manipulator given that the third link should...
I have a three link revolute manipulator at the origin. I know all the link lengths. The joint angle for the third link is coupled to the second such that Theta3 = k*Theta2. I want to determine the joint angles (thetas) of the manipulator given that the third link should lie on a line an angle...
Recently, I bought a book and found a strange question :
In the given figure, PQRS is a quadrilateral. PO and QO are bisectors of angle P and angle Q respectively, then prove that angle QOP = 1/2 (angle R + angle S)
I made several attempts to get the solution but failed. I guess, we need...
Homework Statement
http://184.154.165.18/~devilthe/uploads/1321855187.pngHomework Equations
No idea.The Attempt at a Solution
Alright...so really no idea what to do here...never did any examples like this in class and have scoured over my notes for 2 hours now trying to figure this out...and...
Homework Statement
(z' represent conjugate of complex number z,i is iota =sqrt(-1))
(1)find the locus of z.
|z|2+4z'=0
(2)|z1|=1, |z2|=2, then find the value of |z1-z2|2+|z1+z2|2
(3)z=(k+3)+i[sqrt(3-k2)] for all real k. find locus of z
Homework Equations
|z|2=z.z'
The Attempt at a...
Homework Statement
Show that if M is a surface such that every geodesic is a plane curve, then M is a part of a plane or a sphere.
Homework Equations
- If a geodesic, \alpha, on M is contained in a plane, then \alpha is also a line of curvature.
- Let p be any point on a surface M and...
Hello,
After reading both How to Prove It: A Structured Approach - By Daniel J Velleman, and one of the Lost Feynman Lectures on Planetary Orbits, I'm wondering if anyone could suggest to me any good books they've read (or heard about) pertaining to logic (paired with analysis), or plane...
Hi, recently I've tried to find a good sintetic geometry book with both theory and exercises. Many people said well about this one
https://www.amazon.com/dp/0486458059/?tag=pfamazon01-20
College Geometry. I've bought it and I've got so disappointed. I want a book that has exercises...
Homework Statement
Homework Equations
From my notes: (\psi_{*}v)_{k}(x)=\sum_{i=1}^{n}v_{i}(x)\frac{{{{{\partial}}}}{\psi_{k}(x)}}{{x_{i}}}
The Attempt at a Solution
Okay so i) is fine (ignoring the typo in the question) but I'm a bit confused about ii)
I don't see any need...
The concept of describing something in layman's terms has come into wide use in the English speaking world. To put something in layman's terms is to describe a complex or technical issue using words and terms that the average individual (someone without professional training in the subject area)...
Homework Statement
I am asking about part iv).
[PLAIN]http://img715.imageshack.us/img715/7977/113ivb.jpg
Homework Equations
I guess they would be the ones in the earlier parts...
The Attempt at a Solution
In the given fact, I think x^3 - x - m(x - a) distance from the...
I'm trying to create some kind of demo that rushes through a tunnel.
I have made a random path generator that creates smooth looking paths in 3D space (just some 3D points)
Now I would like to create polies that form a tunnel around that path.
To give an example, let us assume that the path...
Hello , I have something called Asperger's Syndrome and I would like to find a narrow topic or highly specialized field to study in the future as an Aspiring pure mathematician. But I have little or no experience in Mathematics , that's why I'm asking this question. I obsess about geometrical...
when calculating the surface area of curves rotated about axes, you supposedly integrate 2∏f(x)√(1+f '(x)2) dx. This has been explained to me as being derived from the geometry of a frustum, and I understand how the integration works, but I am confused as the process of deriving this formula -...
Homework Statement
Let M be a differentiable manifold. Let X and Y be two vector fields on M, and let t be a tensor field on M. Prove
\mathcal{L}_{[X,Y]}t = \mathcal{L}_X\mathcal{L}_Yt -\mathcal{L}_Y\mathcal{L}_Xt
Homework Equations
All is fair game, though presumably a coordinate-free...
Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.
Homework Statement
Given a triangle ABC with BC = 2AB. D and E are the midpoints of BC and BD respectively. Show that AD bisects angle CAE.
Homework Equations
The Attempt at a Solution
Let AB= x, so DC= AB= x and ED = x/2. If AD bisects angle CAE,
then AC/AE = DC/DE...
The very first time I ever heard about fractals was in my junior year in high school in my Algebra II class when we were studying complex numbers. I was fascinated by these wonderous objects and I've had many questions about them ever since.
Though two of my main questions have always been...
Homework Statement
This is part from a larger construction, but I realized if i can construct this, i can do the larger construction. All ofcourse with ruler and compass.
I have been given an angle with its bisector and a point on that bisector. I have to construct a circle trough that point...
The co-ordinates of a point P on the line 2x - y + 5 = 0 such that |PA - PB| is maximum, where A is (4,-2) and B is (2, -4) will be:
A) (11, 27)
B) (-11, -17) (Answer)
C) (-11, 17) (Eliminated since it does not lie on the given line)
D) (0, 5)
Honest speaking, I don't know what to do in...
Hi,
I am currently studying for a one year postgrad MSc in Theoretical Physics.
In my undergraduate physics degree, my dissertation was on general relativity, so I got a taste of differential geometry and manifolds which I really enjoyed.
At the moment I'm currently attending lectures...