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Questions tagged [geometry]

To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.

3
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0answers
90 views

Differentiation of the law of cosines [migrated]

$$s^2=R^2+r^2-2Rr\cos(\theta)$$ differentiated to give $$2s\text ds=2Rr\sin(\theta)\text d\theta$$ $$\sin\theta\text d\theta=\frac{s\text ds}{Rr}$$ I found this differentiation in a site that was ...
16
votes
11answers
1k views

Is there a proof that the set of real numbers can exactly represent distances? [duplicate]

Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. In physics, we use real numbers to represent distances. For us to be able to do ...
0
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0answers
9 views

What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
4
votes
0answers
93 views

Method to build a polyhedral die with given probabilities [closed]

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
1
vote
1answer
70 views

A problem in newtonian physics [closed]

Hello! I have solved problem 89 using analytic way, i.e, the length of the string will always be constant, so repeated substitutions and differentiating will make way(i got answer as "c"). But is ...
2
votes
1answer
59 views

Can a black hole have a finite perimeter but an infinite radius/diameter?

Recently, I attended a talk at UNC Chapel Hill by Kip Thorne and I recall that he mentioned that black holes can have a finite perimeter but an infinite radius/diameter (since spacetime is curved with ...
2
votes
1answer
38 views

Change in area on a 3-sphere bounded by a trajectory due to a differential change in trajectory

I have a 3 dimensional spherical topology, and I draw a curve onto the sphere labelled by $\vec{n}(\vec{r},t)$. The area bounded by the curve is termed the "Wess Zumino Action" (Hence my motivation to ...
2
votes
2answers
50 views

Why is the angular part of the Schwarzschild metric is not affected by function of $r$ and $t$ as opposed to the first two?

Consider the Schwarzschild metric $$ds^2 = A(r)dt^2 −B(r)dr^2 −C(r)r^2(d\theta^2 +\sin\theta^2d \phi^2),$$ where $A,B,C$ are some functions of $r$. There is no loss of generality if one takes $C = 1$....
1
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1answer
44 views

Definitions of area and volume in $d$-dim spacetime

I am reading Erik Verlinde's paper "Emergent Gravity and the Dark Universe". Equations 2.12 and 2.13 give the area $A(r)$ and volume $V(r)$ respectively. Do $d$ dimensions comprise of both space and ...
0
votes
0answers
21 views

Inside a shell full of strings, what are the components of the network, and what the possible defects?

Imagine that we have a (very) large shell full of holes, and flexible strings go (randomly) from one hole to another. Inside the shell, there will be a network of strings. Can one split this network ...
0
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1answer
39 views

What fraction of events goes through a small area $dA$ a distance $r$ from a spherically symmetric point source as $r\to 0$?

I'm not sure how to use limiting principle to estimate the N fraction when r -> 0 and small area dA stays fixed $ \lim N_{r\to\infty} = \lim\delta A/(4\pi r^2)_{r\to\infty}$ = 1/2 N ? How?
4
votes
2answers
111 views

Number of GPS satellites required to give 100% coverage

I am a Belgian school student in sixth grade of secondary school (the Belgian equivalent of High school). Together with two class mates we were assigned to write some kind of essay about the Global ...
1
vote
2answers
37 views

How is depth perception linked with the separation between our eyes? [duplicate]

I came across a comment online which mentioned that one of the main benefits of having 2 eyes is that we can perceive depth. I tried closing one eye and seeing for myself if somehow everything would ...
10
votes
3answers
997 views

In a ball with random thread/strings, how does the density of threads/strings change with radius?

A large plastic ball full of holes is given. (So the holes are in a plastic shell.) Straight threads connect these holes randomly, by passing through the interior of the ball/shell. For a big ball or ...
-1
votes
1answer
94 views

How to find the center of mass of an homogeneous cube? [closed]

How do I find the center of mass of an homogeneous solid cube of side $L$ analytically? I guess that by side $L$ means that the length of the sides is $L$ and the area is $L^2$, but I'm not sure. I ...
-1
votes
3answers
89 views

What makes displacement a vector?

Displacements are vectors because they add like vectors is the answer. It is also an experimental fact. Though rotations are displacements but not vectors. Is there any more fundamental or intuitive ...
2
votes
1answer
44 views

Why is the idea that two points on a rigid body always correspond to the same distance so important in special relativity?

In Einstein's chapter on physical meaning of geometrical propositions (The special and general theory of relativity) he wrote about supplementing Euclidean geometry with the idea that two points on a ...
0
votes
3answers
61 views

Similar Triangles in $ a=\frac{v^2}{r}$ derivation

Looking at this image, it is said that the two triangles are similar. How are they similar? I can't seem to figure it out. I understand that they're both isosceles triangles, but how can I prove ...
1
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0answers
24 views

Geometrical way to view discretization of energy in quantum mechanics. How commutation relation implies discreteness?

The relation from which discreteness in eigenvalue of the energy of bound state arises is $[x, p]=i\hbar$ followed by the rule that wavefunction should be normalizable. So my question is there a ...
1
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1answer
64 views

Geometric derivation of Lorentz boosts

In two dimensions a very nice parametrization of the rotation group is obtained by the following line of arguments: The group of rotations is connected and compact. Therefore the exponential is ...
0
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0answers
5 views

Calculate thickness of curved membrane from confocal slices

I am imaging a thin, curvy, biological membrane using confocal microscopy. I would like to estimate the thickness of the membrane (assuming it is constant throughout). At every z-slice, I can see the ...
0
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0answers
48 views

Question about the geometric structure of Newtonian mechanics

My point here is about the mathematical structure of Classical pre-relativistic physics and general relativity (GR). It became more clearly, after GR, about the fact that pseudo-riemannian are a nice ...
2
votes
0answers
27 views

What should be the shape of an axisymmetric solar reflector that provides a uniform flux on the surface of a circular cylinder placed on its axis?

I was wondering about creating a solar concentrator with an antisymmetric concave shape that is able to focus the collected light onto a circular cylinder positioned on the concentrator axis. The ...
5
votes
3answers
1k views

When does a vector component keep being a vector, exactly?

English is not my native language, so please forgive my errors. Consider this example: This is a classic: an exercise requiring you to calculate the electric field produced by a charged ring on its ...
0
votes
0answers
37 views

Size of an object (the space it occupies in your whole field of vision) as you move away from it

I'm not sure if this is the right forum for this question, and it must be already answered somewhere so you can just point me to the answer. If there is a disc of radius $r$ and an observer is moving ...
1
vote
1answer
47 views

Why do headlights through a window move around a room?

As I lay in bed this evening a few cars have passed by and my blinds are down. As they come down the street, you can see a sliver of light on the wall on the side of the room that is in the direction ...
2
votes
1answer
123 views

How to calculate the inertia tensor of a spherical cap?

In this question, an attempt is made at calculating the diagonal elements of the inertia tensor of a homogeneous spherical cap, where the $z$-axis is the symmetry axis. The mass moment of inertia ...
3
votes
4answers
165 views

Different expressions for distance & displacement : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$

I came across these expressions in my book. And the book says that all these are different from each other. The expressions are : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$ ...
0
votes
0answers
19 views

Why are vector products defined in such a way? [duplicate]

The dot product of two vectors is an scalar but the cross product is vector, why? What is the physical meaning of these scalar and vector products? Why are they defined in such a way?
0
votes
1answer
56 views

Differential tetrahedron, area of slanted face?

The above diagram shows a cuboid containing a tetrahedron OABC. When the cuboid represents a differential volume element all sides are small in size. With the length of line $OA$ denoted by $|OA|$ ...
-5
votes
4answers
630 views

Is it possible to find the volume of a 4 dimensional object? [closed]

I know there is 4 dimensional objects due to communicating with my friend group. I never found how would you find the volume because it was off the science channel.
0
votes
2answers
38 views

How do I derive a formula for the water level inside a container when it is filled up at a constant flow rate? [closed]

When a container (e.g. Vase, Water Glass etc.) is filled up at a constant flow rate the water height changes differently over time depending on the shape of the container. This video shows some ...
0
votes
1answer
42 views

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates?

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates? Now i did the calculation like this: $\vec R = R \sin\theta \cos\phi \hat{i} + R \sin\theta \sin\phi \hat{j} + R \...
0
votes
2answers
54 views

Deriving density of states in different dimensions in k space

The results for deriving the density of states in different dimensions is as follows: 3D: $g(k)dk = 1/(2\pi)^3 4 \pi k^2 dk$ 2D: $g(k)dk = 1/(2\pi)^2 2 \pi k dk$ 1D: $g(k)dk = 1/(2\pi) 2 dk$ I get ...
2
votes
3answers
111 views

The direction of torque is so confusing [duplicate]

When I was studying cross and dot products, I learned that the cross product of two vectors A and B is perpendicular to both A and B. But my mind is unable to understand that. Since both A and B are ...
3
votes
0answers
34 views

Looking for a geometric cosmology model where time behaves like an orientation

This question is literally inspired from seeing the above scene unfold. Let the merging and splitting light spots you saw in the above gif be pairs of particles and anti particles, let the shape of ...
0
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0answers
44 views

Geometric Surface-to-volume ratios

Is there any physical or scientific significance for the fact that an inscribed sphere in a regular tetrahedron have exactly the same surface to volume ratio at any size?, (i.e. does it have any use ...
1
vote
1answer
49 views

Is there a way to separate 2D from 3D? [closed]

When we see object around us in space, we can always interpret those in 2D, by considering them to pass through a plane, its only when we interact with those objects do we realise that it is 3D, is ...
0
votes
0answers
31 views

Need an equation to calculate the list of latitudes and longitudes of an off-nadir sensor with a given half angle

I need a deterministic way to determine a list of latitude and longitudes of an off-nadir pointing sensor for a given half angle. Below is a depiction of the problem. It isn't too hard to calculate ...
0
votes
2answers
66 views

Unit Vectors in physics

I'm reading the Massachusetts Institute of Technology: "Review of Vectors" , and I've found this: I can't see any relationship between the text that is highlighted in yellow and what's depicted in ...
0
votes
1answer
29 views

Scaling of mass density

Let distances scale as $L'=kL$, $k$ is the scaling factor. Then spherical volume will scale as $V'=k^3 V$. Say, the sphere is filled with mass with constant density. If we scale the model with ...
0
votes
0answers
42 views

Rotor finds equilibrium in TWO positions, why and how?

I have a Mendocino motor, a toy with magnets, solar panels, and coils. Such a thing requires careful balancing (eg. with tiny bits of tape) to achieve sufficient balance that it will start rotating ...
1
vote
1answer
30 views

Computing length of a closed path given a spatial metric

I'm trying to self-study the "cosmology" chapter in Gravitation and Spacetime By Hans C. Ohanian and Remo Ruffini, and I'm stuck on the wording in problem 9.6: The spatial line element for a ...
1
vote
1answer
56 views

Geometrized Algebra and Einstein's Equations

The algebraic properties of the pseudoscalar $i$ follows the ordinary rules for imaginary numbers: So its algebraic properties are ~ $i^2 = -1$ the amazing geometric property is that it is an ...
2
votes
2answers
128 views

Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?

The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
2
votes
0answers
49 views

How to interpret Einstein's development of Euclidean orthogonal transformations in The Meaning of Relativity?

My questions pertain to the development of Euclidean 3-space orthogonal transformations presented in Einstein's The Meaning of Relativity. Also available at https://en.wikisource.org/wiki/...
18
votes
3answers
3k views

Why do water drops form spheres in space? [duplicate]

When water is poured out in space, why does it always take a spherical ball-like shape?
2
votes
0answers
43 views

Is 1/vector is a vector or not? [duplicate]

Let $\vec { A } = a \hat { i } + b \hat { j } + c \hat { k }$. Is $\frac { 1 } { \vec { A } }$ a vector or not, and if it is, then what are its components?"
1
vote
1answer
68 views

How can Young modulus depend on the geometry of the item & Lennard Jones (question related to an older one)

In the question Young's modulus and geometry of test material, the answer was that it cannot play any role. Nonetheless, for the Lennard-Jones solid, in many places the Young modulus is given as : ...
4
votes
1answer
220 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...