OK, that was trivial...I knew it must be easy…if one arranges things the right way. I too was using the pythagorean theorem in order to get rid of some terms. But at the same time I was computing the difference u^2_x+u^2_y-c^2 to get 0…and yeah…somehow fell asleep.
Many thanks, George.
If you mean using the angles in u_x=u \cos \theta,\, u_y=u \sin \theta, \, u'_x=u' \cos \theta',\, u'_y=u' \sin \theta' …does it really help? I tried to plug them in, too…but same thing…the computation gets lengthier and seems to be getting nowhere. There is actually no more on that page, except...
Hello!
Consider the law of addition of velocities for a particle moving in the x-y plane:
u_x=\frac{u'_x+v}{1+u'_xv/c^2},\, u_y=\frac{u'_y}{\gamma(1+u'_xv/c^2)}
In the book by Szekeres on mathematical physics on p.238 it is said that if u'=c, then it follows from the above formulae that...
I think I got now what the message is.
The message is simply that the spatial distance is not a well-defined function on non-simultaneous events, i.e. not independent under the Galilean transforms, for that is what we wish it to be -- the (classical) frame reference change must preserve...
At this stage I actually meant nothing, except that I agreed that what Bill said was in fact exactly what Szekeres said by example, and I pointed to my previous post.
To George:
Aha, ok, if you put it this way then I agree that it becomes a bit less controversial…but still I would then feel inclined to relabel the indices on initial step and then it would actually look all the same, but with relabeled indices. Alright, perhaps I just need a bit more practice...
OK. Now I think that I somehow didn't get the point.
I thought the point is that two non-simultaneous events can be brought by a suitable choice of Galiliean frame to simultaneity, i.e. simply by time shift (adding a constant), so that their distance becomes purely spatial distance.
You...
A Galilean transformation is defined as a transformation that preserves the structure of Galilean space, namely:
1. time intervals;
2. spatial distances between any two simultaneous events;
3. rectilinear motions.
Can anyone give a short argument for the fact that only measuring the...
Hi all! I've got a short question concerning a minor notational issue about tensor contraction I've run across recently.
Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor.
Then their total contraction is zero: C_1^1C_2^2\,A \otimes S=0.
As a proof one simply computes...
Oh my it's getting really funny! Yeah, but the fact is that both sets are in the ball with radius 51, and so their union is bounded. I overlooked the value of the constant K, which should of course be the diameter of the (larger) ball, i.e. 102.
I "repaired" the proof. It should be OK now...
I should have made it more explicit. Here is the complete version:
Let A and B be bounded. Take any point z_0\in X, such that for all x\in A we have d(x,z_0)\ge\sup_{u\in A}d(x,u) and for all y\in B respectively d(y,z_0)\ge\sup_{v\in B}d(y,v). Then with K_A:=\sup_{x\in A}d(x,z_0) and...
Well, I realised this at the very beginning, but it's now that I can write it down:
Let A and B be bounded. W.l.o.g assume that \delta (A)\le\delta (B). By assumption there are numbers K_A and K_B such that \delta(A)\le K_A and \delta(B)\le K_B. Take K:=\max\{K_A,K_B\}<\infty (i.e. we take a...
Suppose \delta (A)<\infty. Let x\in A be arbitrary but fixed. Then d(x,y)\le\sup_{(x,y)\in\{x\}\times A}d(x,y)\le\delta (A). Thus setting r:=\delta (A) we find a (closed) ball \bar{B}_r(x):=\{y\in X:d(x,y)\le r\}. It contains all points of A, since for all x\in A and all y\in\bar{B}_r(x) we have...
I am a bit disturbed by the following elementary observation.
Let (X,d) be a metric space and \emptyset\neq A \subseteq X.
(a) The diameter \delta (A) of A is defined to be \delta (A):=\sup_{(x,y) \in A^2}d(x,y), where A^2:=A \times A
(b) A is called bounded if \delta (A)<\infty.
Now let...
this was easier than I thought. In fact take x=(1,1,...), then 2||x|| is not equal to ||2x||!
First I tried to see whether the triangle inequality is not satisfied, but it did not to work, because both the norm and the metric seem somehow to be "conform" in this respect.
Hello!
It is said that not every metric comes from a norm.
Consider for example a metric defined on all sequences of real numbers with the metric:
d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|}
I can't grasp how can that be.
There is a proof...
An example is
\binom{n+m}{k}=\displaystyle\sum_{i=0}^{k}\binom{n}{i}\binom{m}{k-i}. It is asked to prove it for all naturals n,m,k.
They say it is enough to prove it for example just for m. The reason, which was brought forward, is that in the induction step we assume the truth of the...
Hello!
Question:
if it is asked to prove a statement A(n_1,...,n_k) for all natural numbers n_1,...,n_k, is it actually enough to check its truth by induction on just one of the counters, say n_1?
Hello, could you please check if the reasoning is correct. This is not a homework, just a part of an exercise in a book I'm reading at the moment.
Suppose X is a set, \mathcal{B}:=\{S\subset{}X:\bigcup{}S=X\}, \\...
Hello,
there is a basic lemma in topology, saying that:
Let X be a topological space, and B is a collection of open subsets of X. If every open subset of X satisfies the basis criterion with respect to B (in the sense, that every element x of an open set O is in a basis open set S, contained...
Hi! I'd like to ask the following question.
Does it make sense to take unions and intersections over an empty set?
For instance I came across a definition of a topological space which uses just two axioms:
A topology on a set X is a subset T of the power set of X, which satisfies:
1...
Hello,
Here is a short lemma:
A path-connected space X is simply-connected iff any two loops in X are free homotopic.
My question is whether it is allowed to use a straight-line homotopy straight away in order to construct a free homotopy? For example, let u and v be two loops and w is a...
Hi!
I wonder how to prove that if y(t)=sin(t) solves an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution.
I mean I'm a bit distracted by the fact that all derivatives of y are present here. For example in the equation for a pendulum there are just y and y'' and a...
Hello!
I've got big problems with understanding abstract algebra, the way we deal with it in the seminar on Lie algebras. In just four weeks we progressed up to Levi and Malcev theorems, which are actually the culmination, the say, of classical Lie algebras theory. I didn't think, that the...