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.
In a triangle ABC: A(-2,7) and C(7,-5).
The length of the altitude of AC is 5, and the length of the altitude of BC is the square root of 45. I wish to find the vertex B, given that it is below the line AC.
I need your help, I have no idea how to approach this.
Thank you in advance.
I have...
The Complete Idiot's Guide to Calculus
INTRODUCTION
I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are...
Hello
In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...
Hello.
I am currently working with a beam with the following cross-section:
It consist of three bended sections with the following parameters, alpha = 90 degrees, Thickness = 4 mm, Radius = 50.59 mm.
The top section consist of a small triangle and a rectangle. the triangle have a width = 4 mm...
There are three "concentric" rectangles, one inside the other, like in this figure:
The image is not perfect, but we know that the distance between the sides of the inner-most rectangle and the middle rectangle is always "X", while the distance between the middle rectangle and the outer...
Hello all, sorry for the basic question but I suck at math and it's been many years since I had a course.
Anyway I've got two pieces of metal that bolt together and ideally are square and centered. However the pieces are not exactly square when bolted. The smaller piece is off by about .6mm...
This is a problem I thought of, and I was wondering how to mathematically solve it with an equation.
I tried calling one leg x.
So the other leg, because of Pitagora's theorem, is: √(52 - x2)
The area is equal to the product of the legs divided by two, so:
6 = (x * √(52 - x2))/2
12 = x * √(52...
Assume an eternal, static black hole which has an event horizon, a spherical surface at which any object passes a point of no return and is condemned to move toward a mathematical singularity.
One of the predictions of being inside a black hole is that every spatial direction points towards the...
Homework Statement
If G be the centroid of ΔABC and O be any other point, prove that ,
## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2##
##and,##
##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2##
Homework Equations
i m practising from S L LONEY coordinate geometry first chapter ... only the equation...
If gravity was not geometry.. what conservation law(s) would be broken?
For example.. if gravity was a force.. would other laws of physics be broken?
But gravity as geometry may not be complete answer because it has to be made compatible with quantum. Its quite puzzling.
Homework Statement
The Frenet frame of a curve in R 3 . For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2-sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides the...
Homework Statement
Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'.
(i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc...
Homework Statement
Homework Equations
I could really use a push on how to approach this problem. My primary problem is it asks for the heat flux into the page, which makes no sense to me as that is the z direction and this is in the x/y plane. If anyone could explain this problem and maybe...
An interesting article on the field of Sympletic Geometry built on a shaky foundation and current attempts to fix it:
https://www.quantamagazine.org/20170209-the-fight-to-fix-symplectic-geometry/
Hello,
I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses...
Knowing that Gauss' law states that the closed integral of e * dA = q(enclosed)/e naut, how would you find exactly what A is in any given problem?
I know it varies from situation to situation depending on the geometry of the charge. For instance, I know that for an infinite wire/line of...
1. Here is the Problem:
A line passes through A(2,3) and B(5,7).
Find:
(a) the coordinates of the point P on AB
extended through B to P so that P is twice as far from A as from B;
(b) the coordinates if P is on AB extended through A so that P is twice as far from B as from A.
Homework...
Homework Statement
Let A and B be elements of the line EF such that A=/B prove that the line AB=EFHomework Equations
Axiom that two points determine a unique line and that the intersection of two lines has two distinct points then these lines are the same.
The Attempt at a Solution
[/B]
If A...
Homework Statement
In the drawing you can see a circumference inscribed in the triangle ABC (See the picture in the following link). Calculate the value of X
https://goo.gl/photos/CAacV2dJbUrywfXv92. The attempt at a solution
It seems I found a solution for this exercise with the help of a...
Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor.
But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations.
I am searching for a more concise/abstract...
Homework Statement
Imagine a life guard situated a distance d1 from the water. He sees a swimmer in distress a distance L to his left and distance d2 from the shore. Given that his speed on land and water are v1 and v2 respectively, with v1 > v2, what trajectory should he choose to get to the...
Hey Guys,
through the past years you helped me a lot. But now i have to ask my first question.
I am simulating a solenoid, a very easy one, with a core and a coil. And a translational band (2mm). The only problem is, that i cannot put the band correctly because my solenoid is in a housing, the...
So I always thought that geometry is somewhat different from the rest of math. I mean, most of math is about numbers and relations. While geometry is about space.
Does analysis connect the two? For example, the hypotenuse of a triangle is just a truncated portion of the number line that has...
Hello All,
I am an artist who is just beginning to learn how to think mathematically. Have studied the basics "The Golden Standard", Da Vinci, MC Esher. Given my interest, I was introduced to the work of the Dorothea Rockburne and given this work to critique. While I can do the all the art and...
Homework Statement
So, I have this problem here that's pretty basic, but the solution manual sets different axes, and I'm having a bit of trouble understanding the geometry part, meaning how he applies the given forces to the new axes.
A model airplane of mass 0.750 kg fl ies with a speed...
Homework Statement
[/B]
Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity.
I think I have understood the problem, could you verify my demo please ? Thanks
Homework Equations
The Attempt at...
Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)?
(I used spherical geometry as an example. I would like to know the process in...
Homework Statement
What is the next radius outwards of this Apollonian gasket?
R = radius of outer circle = 5
r1 = radius of largest inner circle = 3
r2 = radius of second largest inner circle = 1
a = unknown radius
Homework Equations
C = 2πr
A = πr2
d = 2r
The Attempt at a Solution
Make a...
Homework Statement
Hartle, Gravity, P9.8
A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M.
(a) What is the period as measured by an observer at infinity?
(b) What is the period as measured by a clock on the spaceship...
Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer):
$$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$
Here R is the radius of the outer circle, r is the radius of an inner circle, x is a function that represents the number of circles in a...
There is a graph showing n on its x-axis and its total stopping time on its y axis.
From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to...
Homework Statement
Let S be a sphere with the equation ##(x-2)^2+y^2+z^2=2 ## and let p a line which satisfies the condition ## p \in (\Pi \cap \Sigma) ## where ##\Pi## and ##\Sigma## are planes with equations:
##\Pi :x+z=2##
##\Sigma: 5x-2z=3##
a) Show that S and p have exactly one common...
Homework Statement
line m2-to-m is 3km longer than line m1-to-m
what are lengths for ##M~~M_2## and ##M~~M_1##
Homework Equations
pythagorean theorem hopefully can be used The Attempt at a Solution
use the picture to your advantage in hopefully creating a valid system of equations. With...
Homework Statement
23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram .
Homework Equations
## (\vec{a}\cdot\vec{b})=0##...
I was in a primary school class room the other day and the teacher asked me for help with this geometry problem, that he had set for his class as an extension challenge, but then realized he couldn't do.
The known angles are marked in degrees. We have to find the angle x.
I spent five minutes...
Okay, I have read on spinors here and there but I really don't understand geometrically or intuitively what it is. Can someone please explain it to me and how/when it is used? Thanks!
Homework Statement
the trajectory γ: ℝ→ℝ3 of a charged particle moving in a uniform magnetic field satisfies the differential equation γ''= B x γ'(t) . where B = (B1, B2, B3) is a constant 3-vector describing the magnetic field, and × denotes the vector product.
(a) Show that the particle...
Dear Physics Forum personnel,
Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential...
I want to build a model of how the ecliptic interacts with the horizon. The horizon appears to be a flat circle, so I thought I'd use a CD sized shape for that. What shape would I use for the ecliptic, and how large relative to the CD? I would like it to be as close as possible to the my...
Reading the English translation of Einstein's seminal paper on GR.
http://einsteinpapers.press.princeton.edu/vol6-trans/90?ajax
This paragraph below on p78 doesn't make much sense to me.
Could you provide a second English translation or even adding math notation.
"Before Maxwell, the laws...
Hello,
My 14 year-old son just started ninth grade. Last year in eighth grade in middle school, he completed a Regents ninth grade level science (living environment) class and its corresponding lab and a Regents ninth grade level advanced algebra class. He passed both Regents exams both of...
I am actually very confused on how to solve the problem. Do I just find the distance between the lines? How do I incorporate the 500 Newtons into the problem? Really confused.
Homework Statement
I've been looking at examples of motion derivations for my class, and it's honestly just very confusing. I heard Dynamics should prep you for this but I must have had a very poor course because we never had to understand geometry and physics to this degree...
Homework...
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...