Space Definition and 1000 Threads

Homework Statement Homework Equations definition of null vector, [/B] The Attempt at a Solution null vector : ## |0 \rangle = (0,0,0) ## inverse of (a,b,c) = ( - a, -b, -c) vector sum of the two vectors of the same form e.g. (c,d,1) + ( e,f,1) = ( c+e, d+f, 2) does not have the same...

For the vector valud function F in the image, the three components of the output vector at a point are functions of (x,y,z)the three coordinates of the point.But while calculating divergence, why is the rate of change of x component of the output along x direction alone is accounted(similarly...

To specify, by fuel I mean fossil, nuclear, chemical and pressurized air. Want to build and design a satellite for my daughter with a camera in it (goPro) with an antenna to be able to view anytime she likes. Want to do a hexagonal approach with solar cells only for power. Want to power the...

I am looking at a couple of very interesting papers, published in MNRAS, that deduce, that the accelerated expansion of the Universe we observe can be attributed to gravitational waves, produced by a very distant merger of two or more universe-mass-scale black holes. The last one is on the...

Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...

I was looking at NASA saying they might not have enough funds to put humans on Mars by 2040. OK, zoom forward maybe 200 years and the cost might be, maybe, just buy a ticket. Or people might already be working there; they will be paid to go there. So, here's the question. What is the least cost...

What's the difference between Euclidean & Cartesian space?

Hello! I am a returning student, 19YO, and will be starting back in community college in a month retaking classes I failed the first go-around with college. This is hopefully less of a "tell me what to do" thread, and more of a "clarify my misconceptions." Basically, I am very interested in a...

as we know light has momentum so theoretically we can use it but is it practical? (also this is it that light only exerts force if incident on something?)

If I consider a tetrahedron of four densely packed spheres of unit radius, what it the radius of the largest sized sphere that can fit in the space in between?

How do you show that there can be only one tangent space at a given point of a manifold? Geometrically it's pretty obvious in 3 dimensions, as one notices that there can be only one tangent plane at a point. But how could we show that using equations?

It seems to me that the concept of a space elevator does not take Coriolis force into account. If the elevator were in built with a space station in geosynchronous orbit and counterweight then there is more to reaching the space station than just climbing the rope. The rope would have to be...

I hear that deepest void of intergalactic space may contain say one particle per cubic cm. I don't want to quibble the amount but let's take that as close enough for my purposes. Now is this figure a statistical average so that if it were correct that each cubic kilometre of deep space would...

Here is a rather interesting NY Times obituary of Constance Adams, space habitat architect. I never heard of her, she died young (53), but sounds quite interesting and did some fun design stuff.

What is the largest possible Rotating wheel space station possible to be constructed with current materials? and what would be the population it would support. also formulas used for calculation.would be useful. Could constructing cylindrical space elevators support more population,

Could someone tell me in what sense the following photo of Hilbert is a infinite dimensional Hilbert Space? It's shown in a pdf I'm reading. Perhaps I'm putting the chariot in front of the horses as one would say here in our country, by considering infinite as infinite dimensional?

Homework Statement From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion: 14.13: Decide if the energy surfaces in phase space are bounded for the following cases: i.) The two-body gravitation problem with E<0 ii.) The two-body gravitation problem viewed from the...

So, I'm investigating a certain way of steering a rocket in space for my first undergraduate research project. Essentially, the idea is to control the location of some mass located on a horizontal track perpendicular to a rocket, so that when the mass is moved, the center of mass of the rocket...

I am reading Feynman's book on QED and something struck me about light. I know that we can only calculate the probability of where a photon goes. After that I came across how a partial reflection affects light. My question is, is there a place in the universe where there is a great thickness of...

Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is...

I was watching a video where well known physicist Lisa Randall said that we still don't know whether space is continuous or discrete. My question is, how do we find whether space is continuous or discrete?? What type of experiments are possible? Is it being done now?? I am thinking this may be...

How long would it take for a blob of mercury (the size of a marble or so) to freeze in space? I'd emagine it would have to boil in some way first, and that would send pieces of it flying around, but the surface tension of mercury is much greater than that of water, or urine (as we'd seen...

Hi PF! Given a function ##B## defined as $$B[f(x)]\equiv f''(x) + f(x) + const.$$ Evidently in order for this function to be in the real Hilbert space ##H## we know $$const. = -\frac{1}{x_1-x_0}\int_{x_0}^{x_1} (f''(x) + f(x))\,dx.$$ Can someone please explain why? I can elaborate further if...

These are from Griffith's: My lecture note says that I am having quite a confusion over here...Does the ##\Psi## in the expression ##\langle f_p|\Psi \rangle## equals to ##\Psi(x,t)##? I understand it as ##\Psi(x,t)## being the component of the position basis to form ##\Psi##, so...

The Dynamical Casimir Effect is the production of real photons from the vacuum in a system where one has moving mirrors (see https://www.technologyreview.com/s/424111/first-observation-of-the-dynamical-casimir-effect/). The frequency of the photons is related to the ratio of the velocity v of...

Help me please:) (1) An idealized engine with constant torque curve means constant acceleration. So for example if there is no air resistance and no frictions and car is on fixed gear and at 3000Rpm it produces 50Hp so at 6000Rpm will produce 100Hp. So he doubles the speed , doubles power and...

I have some questions about space, inertia and frame-dragging that I can’t find anywhere on the internet. The papers that I have found that deal with the subject are often couched in advanced mathematics, so that I can’t get an intuitive grasp of the phenomenon. I would appreciate if...

So for some reason, from time to time, i always come back to this question and i can't remember that part of the physics while i was studying and most of the explanation are pretty generic. Basically how do we know that light actually travels and not just oscillate and transfer energy when...

Homework Statement Have to read a paper and somewhere along the line it claims that for any distinct ## \ket{\phi_{0}}## and ##\ket{\phi_{1}}## we can choose a basis s.t. ## \ket{\phi_{0}}= \cos\frac{\theta}{2}\ket{0} + \sin\frac{\theta}{2}\ket{1}, \hspace{0.5cm} \ket{\phi_{1}}=...

Our home food freezer (ca. 14cu ft) is bigger than we really need. I am planning to fill one shelf with 8-12 gallon jugs of water, with the intent of using its high latent heat of fusion to help keep the contents cold during a power outage. Is this volume actually enough to make a meaningful...

I have a somewhat hypothetical scenario that I've been thinking about lately. I have a pretty basic understanding of physics and the mathematics involved but i enjoy learning about astronomy and astrophysics. So the scenario is this, let's say an object such as a huge star or galaxy was to...

Homework Statement Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V. Homework Equations W = {f ∈ V : f(1) = 1} W = {f ∈ V: f(1) = 0} W = {f ∈ V : ∃f ''(0)} W = {f ∈ V: ∃f ''(x) ∀x ∈ R} The...

Dumb question probably -- But is there a way to measure gravity in a particular area of space, or a "measurement" .. I.E. the gravity 10 miles above Earth v.s 1000 miles above. Not force on another object, but some "unit" or measure of gravity itself.

So I am planning on launching a Satellite to promote the Dogecoin cryptocurrency. One of the main points is printing/painting (Or whatever) the logo on the side of a metal panel. How can I make it so it doesn't melt off or turn white from radiation so quickly?

There seem to be two ways of defning what a vacuum is in QFT: 1. It is state $|0\rangle$ such that $$a_k|0\rangle = 0$$ for all anihilation operators $$a_k$$, with creation operators $$a_k^{\dagger}$$. Thus, it is defined in Fock space. 2. It is state $$|0\rangle$$ such that derivative...

And by how much would the balloon expand, if at all?

Let M = {p, x1, x2, x3, ...} be a metric space with no isolated points. f: M → M is continuous with f(xn) = xn+1, and f(p) = p. We say f separates if ∃ δ > 0, ∋ for any y and z there is some n with |fn(y) - fn(z)| > δ, where fn+1(y) = f(fn(y)). QUESTION: Does f separate?

Hi PF! I'm trying to solve an ODE through the Ritz method, which is to say approximate the solution through a series $$\Phi = \sum_{i=1}^N a_if_i,\\ f_i = \phi_i-d_i.$$ Here ##a_i## are constants to be determined and ##f_i## are prescribed functions, where ##\phi_i## is a function and ##d_i##...

if the Earth were set between 2 Sun (our sun) in the distance which the time passes the exact same as it is right now, does the distance between us and each sun would be the same as it is right now?

Hello! (Wave) I want to find the solution of the following Cauchy problem and determine the space in $\mathbb{R}^2$ where the initial condition defines the solution.$$u_t+xu_x=(x+t)u, u|_{t=0}=\phi(x), x \in [0,1] \cup [2,3].$$ ($\phi(x)$ arbitrary smooth function) I have tried the following...

While studying Relativity I decided to take over a concrete case. So I thought of (what I think is) the simplest case which is the Sphere ##S^2##. So I want to construct the tangent space to the sphere. I think for this I need to embbed it in ##R^3##. I worked out similar problems in the early...

Just wondering how a photon reacts when it is affected by gravity of an object in space like a star. Does the gravity actually bend the light/photons? Say you have a series of photons in a perfect line, all travelling, well, at the speed of light toward a massive object(x). The photons in the...

I'm creating an animation of free precession of a cuboid in GeoGebra. The axis of rotation is not one of the principal axes (but does go through center of mass). Since it's much easier to find the angular velocity and L in the body frame, I defined the e1, e2 and e3 axes (as opposed to the...

As I understand it Georges Lemaître, upon learning about Hubble's discovery that space is expanding, came up with the big bang theory. He thought that if space is getting bigger, then it must have been smaller at one point, and if you go back far enough you get an extremely dense singularity...

Helo. A problem in TOPOLOGY by Munkres states that for a ##T_1## space ##X## countable compactness is equivalent to limit point compactes(somtimes also known as Frechet compactness). Countable compactness means that every contable open covering contains a finite subcollection that covers ##X##...

Homework Statement I have derived the differential equations of a system. They are like the following: a\ddot{\theta} - b\ddot{x} + c \theta = 0 \\ d\ddot{\theta} + e\ddot{x} = F(t) where a,b,c,d,e are constants. I'm having trouble putting it into state space form, since I have the highest...

Hello! I just started reading about ##Z_2## graded vector spaces (and graded vector spaces in general) and I want to make sure I understand from the beginning. So the definition, as I understand it, is that a graded vector space can be decomposed into subspaces of degree 0 and 1. So ##V=V_1...

Hi. I am trying to understand the images that I have posted below. Each layer of Mathematical Structure Hierarchy in the image in this post and Mathematical Space Hierarchy in the image in this post are: statements. 1. What do these statements of each layer of these hierarchies in the...

The question originates from what is often said in accounts of superstring theory, that (perhaps) 7 space dimensions might be curled up into such a tiny scale as to become undetectable. So the question came up, if dimensions might be curled up or extended into different scales, say "very tight"...

For example should a neutron be considered as a point, or does it have volume?