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.
I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
Homework Statement
Find x. L1 and L2 are parallel
choices:
a)2°40'
b)2°30'
c)2°45'
d)2°15'
e)2°20'
Homework Equations
Σleft∠ = Σright∠
The Attempt at a Solution
2x+1+4x-1+4x-1 = 2x-1+2x+4+3x+2
10x-1=7x+5
3x=6
x=2°
Not sure how the answers include minutes '
Maybe I'm overlooking something?
Homework Statement
Find x. L1 and L2 are parallel
Choices:
a)100
b)120
c)140
d)150
e)135
Homework Equations
From the image, the angles of the polygon in blue should satisfy:
6θ + 90 + 4θ + 2θ + 90 + x = 540
12θ + x = 360
x = 360 - 12θ
The Attempt at a Solution
I couldn't figure out how to...
Hello! I just start looking at SDG and I'm already having difficulties with a few concepts as expressed by A Kock as:
"We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R"
"The geometric line can, as soon as one chooses two...
I am reading Spacetime and Geometry by Sean Carroll. In section 1.10 on classical field theory, he uses this formula (1.132)
The curly L is a Lagrange density. S is an action, Φ is a vector potential.
Could the integral also be written as follows?
The equation is $$\|\left(\begin{array} &a\\b\\c\\d\end{array}\right)\|^2=1$$
I was wondering if the number of parameters is 6 and not 3, since we can consider rotations in the differents planes : we choose 2 directions among 4 hence $$C^4_2=6$$ possibilities ?
Hey guys, I would appreciate some help with the math behind creating a working coordinate system for a robotic arm. I am currently trying to determine what servo angles are necessary to align a robotic arm's claw to the given coordinates. Geometrically simplified, the robotic arm is a...
A geometry problem I'm working on has boiled down to finding a function ##f(t)## such that $$f'' + \frac{2}{t}f' + \frac{f'^2}{\left( 1 - \frac{f}{t} \right) t } + \frac{f'f}{\left(1- \frac{f}{t} \right) t^2} = 0$$ It has two fairly simple solutions, namely ##f(t) = a## and ##f(t) =...
<< Mentor Note -- 2 threads merged >>
1. Homework Statement
I need help with question 43 it is attached as an attachment
Homework Equations
#a^2+b^2=c^2#
Sin(a)/A=sin(b)/B
The Attempt at a Solution
Well if I draw a line from the left corner of the bown down to the water it forms a right...
Homework Statement
On the attached two pages from MTW, there are two expressions for the variational principle. I've worked my way through to get to (21.86) (+21.88) and have continued to the bottom of the page to (21.90). I then thought I'd try to use (21.91), (21.92) and (21.93) to see if I...
Let △ABC be a triangle. Let AD and CE be its internal bisectors, with D lying on BC and E lying on AB. Given that ∠CED=18° and ∠ADE=24°, how can I find angles ∠A and ∠C without aid of softwares? Angle ∠B is easy to calculate, as
∠CED+∠ADE=∠CAD+∠ACE=∠A+∠C2=180°−∠B2
42°=180°−∠B2
∠B=96°
The other...
Hi everyone.
I want to build beautiful objects in a computer - object means something which has shape and behavior. I feel comfortable in writing down behavior of an object but I am struggling with writing shape of an object.
A Table:
Wave-forms:
Is it possible to build any shape and any...
Hi, this is my first post with a problem that I have during my Msc Project.
I will briefly discuss my project and the reason why I would like to solve this problem, if you do not want to read this part you can skip it.
I am doing experimental research on the scaled laboratory setup where I...
Why is it that variable geometry nozzles, like those found on jet engine(iris nozzles), are not used as rocket nozzle to provide better altitude compensation?
MATH 4300 - Modern Geometry
(3)
Prerequisite: MATH 3450 with minimum C grade. Topics selected from advanced Euclidean geometry, non-Euclidean geometry, projective geometry. May be repeated once for credit with approval of instructor as subject matter changes.
The above paragraph is the course...
Would need help with the design and material required to punch slots in an AISI 316 stainless steel plate, .039" thick. Slots geometry is .039" width x .500" long. The parts are"scallops" (an oil refinery equipment within distillation towers). Have tried several designs but non of them work...
I know about the construction of the algebra in which operators as in Hilbert spaces are developed from Connes' non-commutative geometry, but I don't find any references [besides further publications by Connes himself] which say that this has turned out to be useful in physics for more than a...
Hello,
does anyone know an (more or less) easy differential geometry book for courses in generall relativity and quantum field theory? I'm looking for a book without proofs that focus on how to do calculations and also gives some geometrical intuition. I already looked at The Geometry of...
In plane geometry it is impossible to construct a line equal to the (cube root of 2) times the length of
a side of a cube, making it impossible to double a cube with a compass and straight edge. Maybe plane geometry needs one more dimension.
What happens if we extend the geometry to 3D(solid...
I would like to ask what I hope are two simple questions about what I recognize to be a complicated subject. I did make an effort to search the Internet for the answers, but the two most promising looking sources I found did not help...
I'm requiring help on a circle geometry question I've done.
The line L, has equation of y=0, and intersects the circle with (3,0) and radius of 29. Find the points of intersection.
My working out:
292 = 841
It's centre is 3,0,
Inserting that in circle equation gives (x-3)2+y2 = 841
Solving...
Hi
I've done a masters taught module in GR and from what I've learned these are two of some of the most important significance of needing a Riemannian Geometry:
1) If we consider the Lagrangian of a freely-falling particle given by ##L= \int ds \sqrt{g_{uv}\dot{dx^u}\dot{dx^v}} ## and find the...
Hello fellows
My background is architecture (bachelor in2016) but for unknown reasons I’ve been fascinated by geometry since last year. it was roughly at the stage where I was trying to grasp ‘the truth ‘ of architecture and somehow got into geometry... happy coincidence.
Since I hadn’t...
Hello,
I've been trying to improve my algebra since I've never been particularly good with math. I'm going through Serge Lang's Basic Mathematics textbook and while I have been learning a lot his proof-based exercises are a pain to get through and the back of the book only provides answers for...
Hello! Currently I own Differential Equations by H.B Phillips, a really old book, but difficult and does it´s purpose. I have only 1 problem, certain exercises require certain geometrical functional study I suppose, for example:
"find the equation of the curves so that the part of every tangent...
My question is not a math question.
I know about the calculus sequence (CAL 1, 2 and 3). I plan to go through all 3 in time. There is no rush for me. However, I know there is a course by the title of Calculus and Analytic Geometry. I want to know when this course is given. Is it given after...
I used to think it was called Zeno's tower, but then realized I probably called it that because it reminded me of his paradox. I have been unable to find this shape on the internet, although I saw a small steel tower outside Stonybrook using this geometry.
I have attached an image of the basic...
I've been trying to wrap my head around equidistant points, like platonic solid vertices inside a sphere where the points touch the sphere surface. This led me to the strange and unusual world of mathematical degeneracy, henagons, dihedrons, and so on, along with the lingering question of...
For anyone who is familiar with the book "Geometry, Topology and Physics" by Nakahara, what do you think are the mathematical and physics prerequisites for this book ?
Mod note: Moved from a technical forum section, so missing the homework template.
@fab13 -- please post homework problems in the appropriate section under Homework & Coursework.
I have the following exercise to solve : I have to find all the points on the surface ##x^2+y^2+z^2=36## (so a sphere...
Homework Statement
[/B]
In a circle with center S, DB is the diameter. The line AC goes 90 degrees from the center point M of the line SB. "
What type of triangle is ACD?
2. Homework Equations The Attempt at a Solution
I can see it is an equilateral triangle, but do not know how to explain...
Hello community
I was hoping someone could help me with the following problem.
I am trying to understand what a middle Ordinate is in terms of geometry (I know it has a versine along a chord).Given the diagram below:-
The Blue line is an arc of some radius.
AB & CB are both tangent to...
I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says,
On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...
This is my college's description for it:
Differential Geometry (3) Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: Calculus IV, and Introduction to Linear Algebra; or consent.
I was doing pretty well in all my...
Homework Statement
Suppose your location has a latitude of 43 degrees and the circumference of the Earth is 40,000 km. Measure the distance from your location along a circle. How far would you have to travel if you go a) to the equator, and b) to the North Pole
Homework EquationsThe Attempt at...
Today, I was at an award ceremony, where in one out of the two scientific lectures, the professor was teaching the basics of Hyperbolic Geometery. However, due to time constraints, he had to teach very fast, and there was no laser pointer, nor a chalkboard, so he couldn't explain very well...
Dear all,
I would need mathematical help to solve for the temperature field in an annular geometry (you find a picture attached below the text):
A copper pipe containing a boiling two-phase flow (in the stratified regime) is immersed in a liquid bath, which temperature ##T_{IY}## is assumed to...
Homework Statement
Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all...
I still have some confusion of the concept of spacetime as geometry. Specifically, what confuses me is causality related to this geometry. My understanding is that stress-energy of matter/energy curves spacetime, the curvature of spacetime tells matter/energy how to move, and dictates to...
https://www2.jpl.nasa.gov/teachers/attachments/parallax.html
It's written that they find the distance by calculating from the parallax angle.
But how do astronomers find the parallax angle?
I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
Prove that tangents to the focal cord of parabola are perpendicular using the reflection property of parabola ( A ray of light striking parallel to the focal plane goes through the focus, and a ray of light going through the focus goes parallel)
I don't know whether this is solvable with just...
My mother's boyfriend's son who is in 6th grade asked me a question about the math they are learning. I thought it wouldn't be too bad but I cannot for the life of me figure it out. A solution would be greatly appreciated.
1. Homework Statement
https://postimg.org/image/1tsa4v4t8r/
Homework...
Hello! I started reading some differential geometry applied in physics (wedge product, Hodge duality etc.) and how you can rewrite classical theories (Hamiltonian Mechanics, Electromagnetism) in a much nicer way. Can someone point me towards some reading about how can more information be...
Could I get recommemdations for textbooks that start by defining cosine and sine using power series and then recover their geometric properties?
John Roe's "Elementary Geometry" does that, but it's not in my current university's library.
Homework Statement
An observer from sea level watches a ship sail away from shore. If the ships height is ##\epsilon \ll 1## after scaling with the radius of the earth, show that the distance the ship disappears will be about ##\sqrt{2\epsilon}##.
Homework Equations
Nothing comes to mind.
The...
What is a symplectic manifold or symplectic geometry? (In intuitive terms please)
I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more...