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.
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...
Author: H.M. Schey
Title: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
Amazon Link: https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20
Prerequisities: Calculus 1,2,3
Table of Contents:
Preface
Introduction, Vector Functions, and Electrostatics
Introduction...
Author: Manfredo Do Carmo
Title: Riemannian Geometry
Amazon link https://www.amazon.com/dp/0817634908/?tag=pfamazon01-20
Prerequisities: Basic differential geometry, topology, calculus 3, linear algebra
Level: Grad
Table of Contents:
Preface
How to use this book
Differentiable...
Hello!
I have done some quantum mechanics, quantum field theory and general relativity. Not much, but enough to say that I have the big picture. Aside from this I have read about analysis on manifolds, functional analysis, Lie algebras and topology.
Now there is a red book in my bookshelf...
Author: Sean M. Carroll
Title: Spacetime and Geometry: An Introduction to General Relativity
Amazon Link: https://www.amazon.com/dp/0805387323/?tag=pfamazon01-20
Download Link: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html
Prerequisities:
Contents:
Contents:
1...
Author: Joe Harris
Title: Algebraic Geometry: A First Course
Amazon Link: https://www.amazon.com/dp/144193099X/?tag=pfamazon01-20
Prerequisities:
Table of Contents:
Preface
Acknowledgments
Using This Book
Examples of Varieties and Maps
Affine and Projective Varieties
A Note...
Hi
I am not familiar with spherical geometry. I am working with elliptical polarization that involves using poincare sphere that present the latitude and longitude angle in spherical geometry. I need to find the great circle angle if given two points that each specified by their longitude angle...
Homework Statement
Of all cylinders inscribed in sphere of radius R largest area of side(M) has cylinder which hight is R\sqrt{2}. Prove.
The Attempt at a Solution
I understand how to prove this i only have problem with derivative:
M=2*r*Pi*H and r=\frac{\sqrt{(2R)^2 - H^2}}{2}...
"use geometry"--Redshift derivation?
1. Use geometry to derive z=v/c where c is the speed of light and z = [v(obs)-v(em)]/v(em).
Homework Equations
None given... Though I am assuming that I am constant.
The Attempt at a Solution
I believe it has to do with the Doppler effect which...
The context of this question is looking at straggling in plasma. I was told there was a simple differential geometry relationship between the following entities:
dE/dx, dE/dy and dE/dt,
where x,y are distance in perpendicular directions (axes on a plane), t is time and I'm using E to...
I want to study algebraic geometry and differential geometry, what should I learn beforehand
what is the relation between abstract algebra and homological algebra:confused:
So I am not asking for any help, on the question, I figured it out and got the right ans. My problem is with the wording.
Question: The points A,B and C lie on Horizontal ground and are such AB=19m, BC=16m and CA=21m
a) calculate the size of angle ACB
The part I have underlined is the...
Hello, I just have a basic geometry question (really within the context of computer graphics). What is the significance in triangulating polygons? Why not squares, or polys with more angles? Why triangles? Is that because it is the simplest representation of a closed area? Also, is it due to...
I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, stoke's thm, dRham chomology, Hodge star operator. Our text is A Comprehensive Introduction to Differential Geometry by Spivak...
My problem is with where the initial height being R - R cos theta. I don't really get where that came from. any hints on where i could look to find the answer or suggestions on how to start are greatly appreciated.
nevermind i figured it out
I have a pretty simple math problem that has been giving me the biggest headache. So what I need to do is optimize the space that I am using. Thus I have a piece that can be seen like a 20 inch in diameter cylinder that is 6 inches tall. I would like to be able to break it into as few pieces as...
I would like someone to give this a quick check, I am really not sure if I am over thinking this question. I got the right ans, just would like a quick check of my method; big thanks in advance.
question: P(-1,5), Q(8,10), R(7,5) & S(x,y) are the veritices of the parallelogram PQRS. Calculate...
I have got next interesting geometry example :-)
I have got regular hexagon ABCDEF, when S (mark for area in Czech... and in the USA, British I don't know :D) 30cm2.
In the hexagon is M.
You know: ABM(S)=3cm2 and BCM(S)=2cm2.
What is S of: CDM, DEM, EFM and FAM?
So, about me... I don't...
Hi all
I work on the rail roads and I am trying to solve a geometry problem which I was hoping someone could help me with.
My problem is this:-
I have a straight rail road. At point A trains can divert onto another road, the other road is curved with a radius (R1), the curvature of the...
Homework Statement
We consider two points, B and I and a line 'a'.
B(0,-4,-7) I(-2,-2,-5) and a: x = y+1 = (z-2)/2
Determine the summits of A and C of triangle ABC knowing that:
-Summit A belongs to the line 'a'
-I is the foot of the height from A (perpendicular to BC)
-The...
Folks,
To date I have been reading about Euler Bernoulli Beam and Timoshenko Beam Theory desribed by the following equations respectively
EBT ##\displaystyle \frac{d^2}{dx^2}\left( EI \frac{d^2 w}{dx^2}\right )+c_fw=q(x)##
Timoshenko ##\displaystyle -\frac{d}{dx} \left[GAK_s...
Homework Statement
Let us assume that ray XY is a limiting parallel to line l, with P*X*Y. Prove that ray PY is a limiting parallel to line l.
Homework Equations
Steps will go something like this: Show that ray PZ meets line lat a point V. Pick a point S such that P is between S and...
Consider the affine transformation \(f(P)=\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}P+\begin{bmatrix}5\\6\end{bmatrix}\).
Find the image of \(ax+by+c=0\) under \(f\).
My answer is \(\left(a-\frac{b}{2}\right)y+\left(\frac{3b}{2} -2a\right)x+4a-\frac{9b}{2}+c=0\).
1. I'm looking for the value of "C". All starting information is in the first attachment.
2. Formulas needed are believed to be Pythagoras theorem and trigonometry.
3. "The attempt at a solution": This would be the second attachment. I've wracked my brain for hours and have come to...
Homework Statement
Please see the attached.
It is a badly drawn sphere :-p
By common sense,the area of the shaded region in the sphere = area of square = r^2
But can anyone show me the mathematical proof?
Moreover,does it apply to the reality?
Imagine when you bend a square sheet with...
So I have been meaning to learn a little algebraic geometry for some time now, but have never gotten around to it. Since classes are just now winding down for the year, I figured that it was an ideal time to self-study a bit.
Now for a bit about my background: I know that commutative algebra...
Let m : [0,L] → ℝ2 be a positively oriented C1 regular Jordan curve parametrized with arc length. Consider the function F : [a,b] x [a,b] → ℝ defined by F(u,v) = (1/2) ||m(u) - m(v)||2
Define a local diameter of m as the line segment between two points p = m(u) and q = m(v) such that:
The...
Homework Statement
Let m : [0,L] --> ℝ2 be a C2 regular closed curve parametrized with arc length, and define, for an integer n > 0 and scalar ε > 2
μ(u) = m(u) + εsin(2nπu/L)Nm(u)
where Nm is the unit normal to m
(1) Determine a maximum ε0 such that μ is a closed regular curve for...
Homework Statement
Consider a function f that can be put in the form f(p) = g(|p|) where g : [0,+∞) -> ℝ is C1 with g(0) < 0 and g'(t) > 0 for all t ≥ 0
Assume that |∇f(p)| = 1 for all p ≠ 0 and prove that the set f(p) = 0 is a circle.
Homework Equations
Given above
The Attempt at a...
I am reading "Mathematical Methods for Scientists and Engineers" by Donald McQuarrie. In his discussion of polar coordinates, he uses a geometric argument to derive the differential area element, which is of course rdrdθ. He shows an isosceles triangle whose two equal sides are r, and the angle...
Homework Statement
Exercise 44 - In the picture attachedHomework Equations
HINT - expand the expression for n and plug the result into equation (70), then use equation (63)
n=(C-B)X(B-A)
n dot (r - A) = 0 (eq. 70)
A dot (B X C) = det {A B C} (eq. 63)...
I've been thinking of a solution, but can't find a one. You have a square of side length 1. You have to draw 2 circles inside the square so they wouldn't go outside the square and at the same time wouldn't cross. What is the maximum area their sum can make?
Homework Statement
Galileo in seeking to discover the laws governing the motion of bodies under the action of their weight, conducted a series of experiments on inclined planes. Choosing as unit of length the distance traveled by the ball the first unit of time, measuring at subsequent time...
Hello everyone!
I just wanted to ask a question about how I should study for differential geometry. Now, as I have it, I've got a few suggestions for books, of which two stand out prominently:
1. John Lee's Introduction to smooth manifolds
2. De Carmo
Which one would be best for self study...
Homework Statement
I am given the following vectors :
p = 3 q = 2 r = 5
2 4 3
-4 -3 -1
They ask to find these:
1. a normal to the plane containing p, q and r.
2. the distance from the origin to the plane...
Homework Statement
#1 Given that the angle between the vectors a and b is 2Pi/3 and |a|=3 and |b|=4 calculate:
(axb)^2 [(2a+b)x(a+2b)]^2
#2 Given three unit vectors, a, b, c such that a+b+c=0 find (a dot b) + (b dot c) + (c dot a)
#3 Given AB=a+2b BC=-4a-b CD= -5a-3b...
Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we...
Geometry Question?
A triangle has vertices, A(1,-1) , B(0,5) and C(-3,0). Find the length of the altitude from A to BC.
You must find the POI of the altitude and BC: that's the hint
So basically i have no idea how to do this...
i got 4.3... but its wrong
Homework Statement
A rectangle is 2 metres longer than it is wide. On the other hand, if each side of the rectangle is increased by 2 metres, then the area increases by 24 square metres. Find the side-lengths of the rectangle.
Homework Equations
The Attempt at a Solution
So my...
I have read many threads concerning many excellent books on calculus but not for analytical geometry. I can't seem to find any thread about good references for analytical geometry. I am taking the course analytical geometry and calculus in college and I need good book in both fields. I have...
Hi friends . is there anybody here who can give me a fortran code for describing fcc lattice and give us the neighbours of each site and also which pay attention to periodic condition ?
thank you for your help my friends.
I am not currently taking a Geometry class, but I've always been interested in Geometry and I'm now looking for a good College-Level Geometry book. All the ones I have checked out from my school's library claim to be College-Level, but mostly talk about subjects I remember learning about in High...
I'm to work out the internal forces in this truss. I know to get the reaction forces first, so I take the moment at the pin support to find the reaction at the roller support. I can't get my head around at how to find x through the geometry to add that force to the moment though. Is there not...
I need help with Part (b). I finished part (a) and attached it as well. My issue comes from how to apply the definition of connection forms to compute them. The definition states: Let E_1, E_2, E_3 be a frame field on R^3. For each tangent vector v at R^3 at the point p let \omega_{ij}(v )=...
Suppose I'm given a random function:
(-8+5 z+4 z^2)\text{}+(7 z+6 z^4-7 z^5)w+(3 z^2-z^3)w^2+(-8 z-2 z^4-2 z^5)w^3+(3-4 z+4 z^2+7 z^3+6 z^4-8 z^5)w^4+(-6 z+4 z^4)w^5=0
Is there no way to determine it's ramification geometry at each singular (critical) point algebraically? I'm pretty sure...
Homework Statement
Consider the points P = (1/2, √3/2) and Q = (1,1). They lie on the half circle of radius one centered at (1,0).
a) Use the deifnition and properites of the hyperbolic distance (and length) to compute dH(P,Q).
b) Compute the coordinates of the images of Pa nd Q...
Hello,
I have been trying to help a friend of mine some descriptive geometry like orthographic projection of planes etc . . . I was wondering if someone can recommend a simple software that has the very basic construction tools like a ruler, compass, protractor etc so as to record some of the...
A quick novice question. What sort of shape would project a holographic reality (Kind of like water spraying from the inside walls of a sphere) from the boundaries of the universe and at the same time contain drains (black holes) that feed the shower?