Geometry Definition and 999 Threads

Consider a 2-sphere on the real plane equipped with the linear map from the sphere to it's equatorial 2-plane by fixing the topmost vertex of the sphere. This is now an analogue of the Riemann sphere in 3-dimensional space, hence we have the "point at infinity" in addition to the usual reals...

hey pf! i am studying fluid mechanics and was wondering if any of you are familiar with a flow around some geometry? for example, perhaps a 2-D fluid flowing around a circle? if so please reply, as i am wondering how to model the navier-stokes equations. i'll be happy to post the equations...

Has anyone considered whether particle entanglement might involve an extra-dimensional substructure of spacetime which negates the need for superluminal communication between entangled particles? If so, what characteristics would such a geometry need to instantly connect particles? Or is it...

I am reading Dummit and Foote: Section 15.2 Radicals and Affine Varieties. On page 678, Proposition 16 reads as follows: (see attachment, page 678) --------------------------------------------------------------------------------------- Proposition 16. Suppose \phi \ : \ V...

I am reading Dummit and Foote: Section 15.2 Radicals and Affine Varieties. On page 678, Proposition 16 reads as follows: (see attachment, page 678) --------------------------------------------------------------------------------------- Proposition 16. Suppose \phi \ : \ V \longrightarrow W...

I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ----------------------------------------------------------------------------------------------------- Definition. A map...

I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ---------------------------------------------------------------------------------------------- Definition. A map...

I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ----------------------------------------------------------------------------------------------------- Definition. A map...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that V is isomorphic to \mathbb{A}^2 and provide an explicit...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that V is isomorphic to \mathbb{A}^2 and provide an explicit...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

In the figure attached, I have to prove AB=2CE given that: line AC is parallel to DE and angles as mentioned in the figure. Can anyone please help me to prove the same?

Hi there, i have a lot of question about Lorentz-Minkowski geometry: 1) Is Lorentz metric degenere or non-degenere? Why? 2) In spacelike subspaces only spacelike vectors live in it there is not problem here but how can we say that timelike subspaces include null and spacelike vectors...

Hi all, I was wondering where I could learn differential geometry online. Preferably via videos. If anyone could post any links to free sites it would be much appreciated. Thanks in advance.

Homework Statement What is the electron pair geometry of the following? SCN− BeCl2 O3 The Attempt at a Solution SCN− Linear BeCl2 Linear O3 Trigonal Planar Homework Statement Would the following structures be polar or nonpolar? (Not applicable if the structure is an...

If I had a sphere with a radius of 100 meters, a diameter of 200 meters, a volume of 4,188,790.20 square meters, and I wanted to place within this sphere a single dot (one dimensional so it doesn't take up any extra space and there is no displacement --if you're thinking in terms of water--)...

Hi, I was wondering if anyone can help me. I don’t have a homework problem, but a problem I have encountered at work. I am a mechanical engineer working in the railway industry and I am struggling with a problem of reconstructing the vertical geometry of a rail in terms of height and...

Now I think there are arguments for the Fine Tuning of the universe. I like Martin Rees' book but I know there are others who disagree with what he said in his book (Just Six Numbers) On another forum I've got involved in a discussion on this topic and I've seen an argument that I think has no...

Hi everyone, I'm an engineer and not a physicist, so forgive me if something sounds stupid. Question: Is it possible to know, which way products of nuclear fission (u-235) will go? Imagine a sheet of single (or few) atom layer crystal of U-235, that is irradiated by neutron source from...

I recently received a private communication that raised this question? I find that I personally cannot imagine the universe having geometry without also having some kind of matter. I mean matter in a general sense, including light, dynamical fields of any sort. No "test particles", no...

A few topics we are covering in class are: Gauss map, Gauss curvature, normal curvature, shape operator, principal curvature. I am having difficulty understanding the concepts of curves on surfaces. For example, this problem: Define the map ##\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2## by...

The school takes place this week, October 7 through 11, hosted by Thomas Thiemann's group. The school is explicitly quantum geometry. Increasingly that is what QG means-- microscopic/high energy geometry of space-time. Focusing on quantizing shape relations rather than on quantum "gravity...

I was watching a documentary about the universe and it claimed that black holes were sometimes as small as 2 kilometers across. Now before this, my general understanding of a black whole was that it had no physical extent in space, that it was just a 1 dimensional singularity, and the black...

So if we imagine a rectangle. The aspect ratio is width/height. What if you want the ratio of the parallel sides? Is there a terminnology for that ratio?

I am reading Spivak, Calculus on manifolds, and I have a basic working knowledge of topology through Mendelson, "Introduction to Topology", I want to learn more about differential geometry, especially co variant derivatives, levi-civita connections, Ricci and Rieman curvature tensors. I know...

Homework Statement A right circular cone is drawn above, with 2 circles centered at A on its base as shown. AB is the height of the cone, the measure of <ABC is 60° and BC has a length of y. If BD bisects <ABC, which one of the following gives the area of the smaller circle in terms of y...

Homework Statement LPN is a tangent to circle ADP. Circle BCP touches the larger circle internally at P. Chord AD cuts the smaller circle at B and C and BP and CP are joined Homework Equations The Attempt at a Solution ∠P4+5 = ∠B1 (tan chord theorem) ∠P1+2 = C1 (tan chord theorem)

Homework Statement The planes ax+by+cz=1 meets the axes OX, OY, OZ in A,B,C. A plane through the x-axis bisects the angle A of the triangle ABC. Similarly, planes through the other two axes bisect the angles B and C. Find the equation of the line of intersection of these planes. Homework...

I've been wanting to see what this topic is all about for awhile now. I see the word "Manifold" and other terminology floating around on the forum. It got me really curious. I wonder what the prerequisites are to reading a book like this? Hypothetically I have the prerequisites, what would...

Hello, Please forgive me if it's in the wrong sub-forum because don't know where to place it. I need help solving this problem it's chapter 1 and in our class we are already in chapter 5 so I might sound like a fool asking the teacher about it, I was revising and decided to do some questions...

Hi! I'm trying to read up on the subject of hypersurfaces related to GR; First and second fundamental form, Theorema Egregium etc.. Does anyone know any good treatments? (Books or notes)

Gentlemen, the the image shows 2 N48 neodymium magnets of the same volume 54 in2. On several website I can use magnetic flux density calculators to get the Br of the rectangular magnetic but haven't found one that calculates the value of the arc magnet. The Br value of the rectangular magnet is...

Hello MHB. My masters thesis is on the subject of tensegreties (Most likely you don't know what that means.) Their study requires familiarity with Algebraic Geometry. I have no background in AG. I need to understand this paper: https://docs.google.com/file/d/0B77QF0wgZJZ7d2g2S3g1RVlQOE0/edit...

Prove that in any triangle, if the angle bisectors of two angles are congruent, then the triangle is isosceles Before I give my proof, here is a lemma to it: If a pair of vertical angles both have angle bisectors, then all resulting angles are congruent. Given: Vertical Angles ∠2 and ∠4, and...

The curve in the figures above is the parabola $y=x^2$. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. Five normal lines are shown in the figures above. For a while, the $x$-coordinate of the second quadrant...

Homework Statement Calculate the cross product of (3u+4w)xw assuming that uxv=<1,1,0>, uxw=<0,3,1), vxw<2,-1,-1)Homework Equations Possible Relevant eqation: i) wxv=-vxw ii)vxv=0 iii)vxw=0 if and only w= λv for scalar λ or v=0 iV)(λv)xw=vx(λw)=λ(vxw) V) (u+v)xw= uxw+vxw ux(u+w)=uxv+uxw The...

Hello, first post, don't know if I got the section right tough.. Basically I was wondering, if the efficiency of radiant barrier (for home insulation or industrial, high T) could be improved with different surface geometry eg wavy, triangular, pyramids? Why I'm wondering - the reflective...

Homework Statement Solve for \theta Homework Equations Unknown The Attempt at a Solution I know, but I have forgotten my basic geometry.

Are "Commutative Algebra" and "Algebraic Geometry" useful for physics? Hello, I'm considering taking Commutative Algebra, and perhaps even Algebraic Geometry (for which the previous is a prerequisite). In the first place I would take it for the enjoyment of mathematics and to give me an...

Author: Tom Apostol, Mamikon Mnatsakanian Title: New Horizons in Geometry Amazon Link: https://www.amazon.com/dp/088385354X/?tag=pfamazon01-20

Homework Statement A triangle ABC is given where vertex A is (1,1) and the orthocentre is (2,4). Also sides AB and BC are members of the family of lines ax+by+c=0 where a,b,c are in Arithmetic Progression. Find vertex B. Homework Equations The Attempt at a Solution I can write the equation of...

Geometry -- calculating a distance Homework Statement I'm trying to calculate the distance from the center of one of these three tangent circles to the center of the entire shape (that is, the center of the "curvy triangular" region between the circles). Each circle has radius R. Here is a...

I would like to learn DG so I picked up "Differential Geometry" by Erwin Kreyszig. I'm finding it too difficult to understand, especially the notation. What books on DG would you guys recommend to beginners?

Homework Statement Neutrons are emitted uniformly from the inner surface of a thin spherical shell of radius R at a velocity V. They are emitted normal to the inner surface and fly radially across the volume of the sphere to be absorbed at diametrically opposed points. The neutrons are non...

I simply cannot get interested in geometry, but i realize it's an adequate skill to have, does anyone have any tips for me? It's also not a matter of mathematical skill since I'm still in the 10th grade and am already capable of doing S.A.T level 5 problems, i just find them a nuisance to...

Hello everybody, Based of some information that I recently learnt(which I don't know if they are right or wrong), I start asking myself this question about the euclidean geometry. Ok, this geometry is basically founded on straight lines, and what I have learned is there is no such a thing as a...

Consider a pyramid whose base is an $n$-gon with side length $s$, and whose height is $h$. What is the radius of the largest sphere that will fit entirely within the pyramid?

Please move this if it's not the appropriate section of the forum. Could anyone point me towards some good supplements for self studying geometry? I just finished trigonometry, and I'm starting calculus, and algebra/trig based physics in a few weeks once fall courses start. However, I've...

Suggestions for Analytic Geometry books; What do you have in mind?:)