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.
Universe geometry article simpify?
article development for the Forum on geometry suggestions, as well as any errors etc are welcome
particularly on how to keep the FLRW metrics but simplify the explanation...
Universe geometry
The origins of the universe is unknown in cosmology. The hot...
Here's the introduction of the paper by Freidel and Hnybida. Quantum geometry is built up of chunks of geometry that contain information relating to volume, areas, angles made with neighbor chunks, etc. The Hilbert space that these chunks (called intertwiners) live in needs a set of basis...
Is it necessary to finish Spivak's little book to move on to Spivak's Differential Geometry I, or is the material on differential forms and integration on manifolds in Chapter's 4 and 5 of Spivak's little book covered in Differential Geometry I?
Homework Statement
Calculate the volume of the body that is bounded by the planes:
x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2
Homework Equations
The Attempt at a Solution
I made a variable substitution
u = y+z
v = y-z
w = x
which gave me the new boundaries
u+w = 2...
I've been developing an article on universe geometry that hopefully forum members will find as a useful reference,and would like some assistance in examining the accuracy, means of simplifying and details forum members would like added.
The article is on a personal website that references...
Hello,
I am new very new in this subject. I have a curiosity in understanding diff.geometry. I have some questions (which might sound elementary) to ask:
(1) Is diff.geometry a subject related to the study of surface, curvatures, manifolds?
(2) How it is different from Euclidean geometry...
A polygon with nonnegative area cannot be formed with fewer than 3 points.
A polyhedra with nonnegative volume cannot be formed with fewer than 4 points.
A hyperspace with nonnegative measure cannot be formed with fewer than n points.
What I mean by "3 points" is that the cardinality of the set...
Hello,
I am a beginner. I am self taught in differential calculus. Can you please suggest me any book, as a beginner, to have a very basic idea and overview on Differential Calculus.
Any free e-book?
Kindly suggest.
Hello,
Analytic geometry has provided us with such profound tools for thinking that it is hard to imagine what thinking must have been like before we had such tools. Two particular developers of these tools are Pierre de Fermat and Renee Descartes in 17th century France.
I would like to...
I could not post this to the resource forums, so I am posting it here.
I am looking for a Geometry textbook for pre-service teachers. The text ideally should incorporate some constructivist practices and the use of technology to help visualize geometry problems. Most of the teachers will be...
Homework Statement
Solve for the flux distribution using the 1D neutron diffusion equation in a finite sphere for a uniformly distributed source emitting S0 neutrons/cc-sec.
My problem right now is that I can't figure out the boundary conditions for this problem. We usually work with...
Homework Statement
ds^2 = g_{tt} dt^2 + g_{tx} (dt dx + dx dt)
with g_{tt} = -x and g_{tx} = 3
"Show that this is indeed a spacetime, in the sense that at every point, in any coordinates, the matrix g_{\mu \nu} can be diagonalized with one positive and one negative entry. Hint: You...
Hello. I can't seem to wrap my head around the geometry of the gradient vector in ℝ3
So for F=f(x(t),y(t)), \frac{dF}{dt}=\frac{dF}{dx}\frac{dx}{dt}+\frac{dF}{dy}\frac{dy}{dt}
This just boils down to
\frac{dF}{dt}=∇F \cdot v
Along a level set, the dot product of the gradient vector and...
Homework Statement
i'm confused as to why a molecule with 3 bonding pairs and 2 lone pairs takes on a t-shape rather than a trigonal planar shape.
My notes say that this is because in a t-shape, there are less 90 degree angles between the lone pairs and the bonding pairs than in a...
Homework Statement
Find parameter a so that line y=ax + 11 touches ellipse 3x^2 + 2y^2 = 11
The Attempt at a Solution|
I can rewrite ellipse equation like \frac{x^2}{\frac{11}{3}} + \frac{y^2}{\frac{11}{2}} = 1
And i know that line y=kx + n touches ellipse when a^2k^2 + b^2 = n^2...
My geometry is pretty weak and I want to strengthen it.. because the other day my math teacher asked me what a tetrahedron was and I didn't know ...
I've been desperately looking for this book "Geometry for the Practical Man" by J.E. Thompson.
I have all the other books in the series and...
These are some of the factors I brainstormed that goes into the functionality of a spotlight:
1) Light output of the light source (measured in luminous flux, if I recall correctly)
2) The reflective potential of the reflectors surrounding the light source
3) The geometry of the reflectors...
Hi Friends,
I am getting problem in a geometry problem. Please help me to find the answer.
The problem is as follows:
AB, BC, CD, AD are the tangent of circle of radius 10 cm. and center O. If the length of BC = 38 cm and CD = 27 cm. Then find the length of AB. Here tangent AB and AD are...
Hello All,
I have been give a particular task with packing hexagonal shapes with radius 0.105m, into different circular areas. This is not a 3D problem, and I have been trying to search for answers on the topic of "packing" but haven't seemed to find any that fit my requirements.
So the idea...
Let sharp triangle ABC inscribed circle $(O;R)$ and $H$ is orthocenter of triangle ABC. circle $(E;r)$ tangent to $HB$, $HC$ and tangent to in circle $(O;R)$.
Prove that: midpoint of $HE$ is center of the circle inscribed the triangle $HBC$
Hi, I'm a Physics undergraduate, and this semester I have the option to choose between Geometry (Axiomatic Euclidean Geometry) and other disciplines. In the next year I want to be ready to study Differential Geometry, but I don't know if I need to study Euclidean Geometry first. The teacher of...
Homework Statement
In the upper half of a (x,y) plane endowed with a refractive index of n(y) = 1/y, find the form of light ray.
Homework Equations
l = ∫n dl
The Attempt at a Solution
My method is to construct a functional for optical path, obtaining the result using...
Homework Statement
Let α(t) be a regular, parametrized curve in the xy plane viewed as a subset of ℝ^3. Let p be a fixed point not on the curve. Let u be a fixed vector. Let θ(t) be the angle that α(t)-p makes with the direction u. Prove that:
θ'(t)=||α'(t) X (α(t)-p)||/(||(α(t)-p)||)^2...
I heard that some physicists are trying to determine the spacial/geometric curvature of the universe by measuring the angles of distant stars (a very large triangle).
Is this possible? Or is Poincare correct when he said that there is no preferred geometry and that there is no experiment...
Homework Statement
I need to derive an expression for the displacement of light as a function of thickness of glass and the angles.
I will post a screen shot of the formula to be derived but it can also be found here...
Geometry Question -- Modeling bolt with flange end
Homework Statement
Hi
I attached a picture of bolt call ed hollo bolt, in which the sleeves open when the cone pushed inside,, if i Know all the dimensions,
I ask an expert and he tell me that
However, the
difficulty is...
We recently had a long thread https://www.physicsforums.com/showthread.php?t=666861 about cases where raising and lowering indices isn't completely natural, i.e., where a vector "naturally" wants to be upper-index or lower-index.
If you have a metric, then it's pretty clear to me what...
Hi, I read in Padmanabhan's book that \nabla_a J^a=0 implies that there exists an antisymetric tensor P such that J^a= \nabla_b P^{ba}. What's the name of the theorem? Any reference?
Thanks
Suppose x(t) is a curve in ℝ^2 satisfying x*x'=0 where * is the dot product. Show that x(t) is a circle.
The hint says find the derivative of ||x(t)||^2 which is zero and doesn't tell me much.
I was hoping for x*x= r, r a constant.
Homework Statement
What is the area of a right triangle whose inscribed circle has radius 3 and whose circumscribed circle has a radius 8?
Homework Equations
The diameter must be the hypotenuse of the circle
The Attempt at a Solution
The answer is 57, but I do not know the...
Hi there. I want to learn some differential geometry on my own, when I find some time. My intention is to learn the maths, so then I can get some insight, and go more deeply on the foundations of mechanics. I need to start on the basics. I had some notions on topology when I did my analysis II...
Hi
I am struggling to see how the relation:
(df:U)(dh:V)-(dh:V)(df:V) (1)
can be rewritten as:
U(fdh:v)-V(fdh:U)-fdh:[U,V] (2)
I have tried expanding (1) out by making a scalar vector product of Udh:V but i don't think that is right and i am just not sure how best to proceed. It was...
We know that the spacetime of General Relativity with a single electron in otherwise empty space is hardly curved, basically zero.
In Kaluza–Klein theory with a single electron in otherwise empty space is there a type curvature due to the charge of a single electron?
Is the amount of...
Here's a nice geometry problem, not hard at all if you can see what's really going on.
https://lh3.googleusercontent.com/-qqF-3y81rvI/UQjt7jishKI/AAAAAAAAAII/gP8G0dgKCmc/w497-h373/cercles.gif
Solution (don't click if you want to work it out yourself!):
the differential geometry is so abstract to understand. All are terms and theorem. How to understand it? can someone give me some method and guidance to learn it. HELP!
Author: Serge Lang
Title: Fundamentals of Differential Geometry
Amazon Link: https://www.amazon.com/dp/038798593X/?tag=pfamazon01-20
Prerequisities: Grad Analysis, Differential Geometry
Level: Grad
Table of Contents:
Foreword
Acknowledgments
General Differential Theory
Differential...
Author: Serge Lang and Gene Murrow
Title: Geometry
Amazon Link: https://www.amazon.com/dp/1441930841/?tag=pfamazon01-20
Prerequisities: High-school algebra
Level: High school
Table of Contents:
Introduction
Distance and Angles
Lines
Distance
Angles
Proofs
Right Angles and...
Author: William Burke
Title: Applied Differential Geometry
Amazon Link: https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20
Prerequisities:
Level: Undergrad
Table of Contents:
Preface
Glossary of notation
Introduction
Tensor in linear spaces
Linear and affine spaces...
Author: H. S. M. Coxeter, Samuel L. Greitzer
Title: Geometry Revisited
Amazon Link: https://www.amazon.com/dp/0883856190/?tag=pfamazon01-20
Prerequisities: High-School mathematics
Level: Undergrad
Table of Contents:
Preface
Points and Lines Connected with a Triangle
The extended Law...
Author: Michael Spivak
Title: A Comprehensive Introduction to Differential Geometry
Amazon Link:
https://www.amazon.com/dp/0914098705/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098721/?tag=pfamazon01-20...
Author: Robin Hartshorne
Title: Geometry: Euclid and Beyond
Amazon Link: https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20
Prerequisities: Proofs, intro to abstract algebra
Level: Undergrad
Table of Contents:
Euclid's Geometry
A First Look at Euclid's Elements
Ruler and...
Author: C.J. Isham
Title: Modern Differential Geometry for Physicists
Amazon Link: https://www.amazon.com/dp/9810235623/?tag=pfamazon01-20
Table of Contents:
An Introduction to Topology
Preliminary Remarks
Remarks on differential geometry
Remarks on topology
Metric Spaces
The...