Yes. And I want to understand how is it possible that time coordinate assigned to that event is different for the train passenger and the observer. What is reason of this thing?
Yes. I used this term wrong. I don't mean about "time simultaneous" or "at the same time". I mean that when light reach to end of train with point of view of passenger then light reach to end of train with point of view of observer. Is it true?
Is it true that in previous example light reach left end of train with point of view of passenger and with point of view of outside observer simultaneously?
Another example is the propagation of light from the middle of a moving train to its left and right
ends. From the train passenger's point of view, the light will simultaneously reach the right and left ends of the train, with point of view of an outside observer - at different times.
In this...
I mean another. I mean that in moving frame and in non-moving frame same event happens simultaneously.
Ok. For example, light clock. Light reach to end point simultaneously in non-moving and moving frames.
https://en.wikipedia.org/wiki/Time_dilation#Simple_inference
For example, light clock and mechanical clocks. We can simultaneously start mechanical clocks at initial point and if light reach to end point we can stop simultaneously our mechanical clocks in different frames.
Hello!
I try to understand how in different frames clocks tick and stop simultaneously but show different time? I suppose that velocity is reason of time dilation effect but I'm not sure.
Thanks.
Thanks to all! My mistake was that I tried to find time of motion in moving frame but distance in this frame is 0 because clock in moving frame is at rest.
Ok. I'll try to rephrase.
Let's write equation for non-moving reference frame:
##(cΔt)^2 = (cΔt0)^2 + (vΔt)^2##
How to write equation for moving reference frame?
I wrote that time is Δt0, is it true? But I'm confused because if we're moving from A to B then our distance is vΔt for our reference frame. And if time is Δt0 then distance is vΔt0. It's wrong result.
I considered example of time dilation with light clock. I have a question about measuring time in reference frame with clock.
If we know that clock move from A to B in the reference frame with clock then what time of motion is measured in this reference frame? (In non-moving reference frame...
Hello.
I read about smooth infinitesimal analysis and I have several questions:
1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6)
2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...
Hello.
How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.)
Thanks.
I read in this source: http://www.bndhep.net/Lab/Math/Calculus.htm
The fact that x^2 becomes insignificant compared to x for very small values of x is a fundamental principle of infinitesimal calculus. We say x is infinitesimal when we allow its value to approach zero, but never actually reach...
What is relation between ##\Delta x## and ##dx##? If we want to get term with dx then we discard ##\Delta x^2## and other high-order terms in expression. Is it true?
Hello.
As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##.
I read that infinitesimals have property: ##dx+dx^2=dx##
I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...
Ok. We have 2 similar expressions: 1. ##2x \Delta x + \Delta x^2## where ##\Delta x## is variable and 2. ##x+x^2## where ##x## is variable.
In 1. case we'll get ##2xdx## when ## \Delta x## tends to zero.
In 2. case we'll get ##x## when ##x## tends to zero.
But in the first case we'll get...
Hello.
Let's assume that we have ##2x \Delta x + \Delta x^2##. When ##\Delta x## tends to zero we can neglect ##\Delta x^2## and we'll get ##2xdx##.
Let's assume that we have ##x + x^2##. When ##x## tends to zero we can neglect ##x^2##. Will we get an infinitesimal ##x## as such as ##dx##?
Thanks.
Hello, @fresh_42
You wrote that "In the first case ##dr \approx \Delta \, r## is an infinitesimal small change in ##r## and in the second it is an infinitesimal small piece of ##r##, so ##dr \approx h##."
I have difficulty understanding this topic.
How is it possible that ##dr \approx \Delta...
Hello.
There are 4 types of infinitesimals:
1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers)
2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number.
3) Surreal numbers: { 0, 1...
Hello!
As is known, \Delta y = dy for infinitesimally small dx. It's true.
But if we have graph we may see that \Delta y isn't equal to dy even for infinitesimally small dx. Why is that so?
Thanks!