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
The global topology of a ##2+1##-dimensional universe is of the form ##T^{2}\times R_{+}##, where ##T^{2}## is a two-dimensional torus and ##R_{+}## is the non-compact temporal direction. What is the Fermi energy for a system of spin-##\frac{1}{2}## particles in this...
Homework Statement
Given the diagram below, showing the path of a geocentric satellite S flying over a ground station G, find an expression for the geocentric semi-angle ##\phi## in terms of ##\epsilon##, the radius of the Earth ##R_E##, and the height of the orbit ##h##.
Homework Equations...
This is a programming issue, but the question is really about geometry. Let me 'splain.
I am making a page control in HTML/CSS/JS *no wait - come back! I swear it's not about programming!* that allows the user to play with the opacities of three overlapping images (eg. imageA = opacity 60...
Homework Statement
Predict the hybridization and geometry of the indicated atom. Answer all parts.
Homework Equations
CH3CH2
There is a ( - ) sign above the second C
Hybridization = ?
Geometry = ?
The Attempt at a Solution
I know I'm supposed to count the groups around the atom, but I'm...
I would like to begin my first exploration of the arts of differential geometry/topology with the first volume of M. Spivak's five-volume set in the different geometry. Is a thorough understanding of vector calculus must before reading his book? I read neither of his Calculus nor Calculus on...
How are those two geometries realeted?
Conformal geometry is a metric geometry. Projective geometry is not. But the stereographic projection is related to the conformal geometry.
Or does someone know a book/ notes where the individual geometries (affine, projective, euclidean, hyperbolic...
Simple geometry can compute the height of an object from the object's shadow length and shadow angle using the formula: tan(angleElevation) = treeHeight / shadowLength. Given the shadow length and angle of elevation, compute the tree height.
What I have so far:
import java.util.Scanner...
So, I'm reading a physics book, and it talks about Newton's three laws, of course, but then after that it says that if a force of f pushes on an object at angle Θ, then the force in the x direction is f ⋅ cos(Θ), and the force in the y direction is f ⋅ sin(Θ).
Where did THAT come from? Do we...
How can i use F4 tally in hexagonal lattice geometry which composed of circular fuel assembly? For example one hexagonal lattice includes circle which is cell number 5 and another hexagonal lattice includes circle which is cell number 7 etc. I want to use all these cells in one f4 tally.
Please...
So I have a rectangular section as my geometry that needs to be formed in three stages. So after 1st operation I want to take that deformed shape and perform another operation (all analysis use Explicit Dynamics). Is this possible in ANSYS?
Thanks in advance
A problem on geometry proof
Hi (Smile),
When considering the \triangle ABM E is the midpoint of AB
& EO //OM (given).I think this is the way to tell AO=OM , Help .Many Thanks (Smile)
Hi, there is a book of dg of surfaces that is also about tensor calculus ?
Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus!
Thank you in advance
Hi,I have been stuck on this problem
The midpoints of the sides AB and AC of the triangle ABC are P and Q respectively. BQ produced
and the straight line through A drawn parallel to PQ meet at R. Draw a figure with this information
marked on it and prove that, area of ABCR = 8 x area of APQ.
I...
The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as:
$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}
$$
My question is that it seems that...
Hi guys,
I'm thinking of maybe of studying differential geometry as part of my undergraduate degree. However, it's not for physicists, it's a full on formal mathematics course specifically for mathematicians. I'm not sure whether it's a bit overkill and won't actually be useful. We don't have a...
Hello from Italy
I'm switching from CS to Physics BS because i personally find it more various and interesting (and in Rome there is one of the best physics school in the world).
Mathematical analysis is a common subject and my credits will be recognised but i didn't study Linear algebra yet...
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused...
Projective Algebraic Geometry - the Projective Plane ... Cox et al - Section 8.1, Exs 5(a) & 5(b)
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra...
I'm trying to self-study general relativity, but I encounter a problem. I can easily understand the ideas and the results, but there are some things related to 4-d geometry that I can't find, like what are Christoffel symbols and how to solve the equation of a geodesic line. I searched for those...
As the title implies, I'm looking for books on non-euclidean geometry. I'm not looking for very advanced thing, more on some book with a good introduction to this topic.
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on...
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on...
I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information?
The Riemann tensor contains all the informations about your space.
Riemann tensor appears when you compare the change of the sabe...
This is not a homework question. School year has ended for me and I'm doing some revision on my own.
I want to proof the following because in an exercise I had to find the equation of the line that passed through a given point and 2 given lines.
If a line r intersects with 2 given crossing...
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on Chapter...
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on Chapter 8...
hi.. i want to import only the geometry defined in mechanical apdl into ansys workbench so that i want to have the freedom to change the geometry and meshing whenever i want to. thankyou
Hello, I was wondering if anyone knows about the the book
"The Geometry of Spacetime: An Introduction to Special and General Relativity" by Callahan
and what their opinions are.
Thanks!
Hello,
I would like to know if anybody here has used the book "Differential Forms and the Geometry of General Relativity" by Tevian Dray and how they found it.
Thanks!
Can someone help me with this?
(dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections).
A=pir^2
(dA/dt)=2pir(dR/dt)
Multiply through by (1/2pir)
(dA/dt)/(2pir)=dR/Dt
What is the rate of change of the radius for a circumfrance of 2
I just used the related rates formula that I derived for...
I have found a way to construct a pentagon using only the grid system. The internal angles are all within 99% accuracy and all the angles to the 3rd decimal add up exactly to 540 degrees. This is without the use of a compass and bearing in mind that the internal lines intersect at the ration of...
I recently decided to take a whack at this problem. Came up with an interesting approach, thought it would make a good conversation topic.
Anyone else tried to do this? What were your results?
Homework Statement
let be ABC a generic triangle, build on each side of the triangle an equilater triangle, proof that the triangle having as vertices the centers of the equilaters triangles is equilater
Homework Equations
sum of internal angles in a triangle is 180, rules about congruency in...
So I'm reading the Schaum's outlines while trying to prepare for a big test I have in September. And I'm trying to understand something here that maybe someone can offer some clarification and guidance.
So, using Coulomb's Law, we can find the electric field as follows:
\begin{equation}
dE...
It's been a long time since I posted a riddle, and probably nobody missed them. Anyway, here is a collection of 3 well known geometry problems with different difficulty levels. The only reason there is an "I" prefix is because of the third part of the problem, which is a bit harder. The first...
[Moderator's note: this thread is spun off from another thread since it was a subthread dealing with a separate topic.]
There is definitely a maximally extended spacetime but there is no maximally extended spacelike surface of constant Schwarzschild coordinate time t. The spatial curvature...
Hello everyone, I've 2 books on manifolds theory in e-form:
1) Spivack, calculus on manifold
2) Munkres, analysis on manifold
What would be good to begin with? :oldconfused:
Thank you in advance
Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$.
Its volume distortion is defined as $G=det(DftDf).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$.
What happens for $n>1$? Can one bound from below this $G$? If...
HI , i am having a problem in reinmann geometry as i am not able to cope up with the language used in
"introduction to reinmann geometry-with applications in mechanics ang relativity".Can anyone suggest an easy to go with book, for a beginner for self study
hi, Initially, I know how to take surface integral of möbius band via given parameterization, but I really wonder how these parameters are created. How can we derive these parameters ?? What is the logic of deriving such a good parameters?? Could you give some proofs??
It is said that curves of the second order which we usually refer to as ellipse, parabola and hyperbola, i. e. conics, are all represented on projective plane by closed curves (oval curve), which means there is no distinction between them. Why is it?
Projective space can, in principle, be...
I made a picture because I'd struggle to get out a question without it. In the picture all things are constant except the strength of the magnetic field. It is at two different values. We see 2 cycloid paths of electrons that starts at rest. The circumference of the large path is exactly twice...
Homework Statement
Say we have two manifolds N(dim d) and M(dim d-1). Let Φ: M →N be a diffeomorphism where Σ = Φ[M] is hypersurface in N. Let n be unit normal field (say timelike) on Σ and ⊥ projector (in N) defined by:
⊥^a_b = \delta^a_b + n^a n_b
Where acting on (s, 0) tensor projection...
So I was reading now about the new geometries and I wanted to know if I can study the Reimann Geometry knowing that I finished high school or if I could just know about it but not about the formulas. I am so interested in the subject because it is used in astronomy.