studiot

Hello Tony, before I answer this can you tell me if you have done a cooling or melting curve experiment? That is plotted the Temperture v Time graph of some ice or wax as it melts and then warms up or some water or oil as it cools and solidifies?

Well one way to solve a pair of simultaneous equations is by substitution. This means using the second equation to find one unknown in terms of the other and then replacing it in the first equation. This will obtain a single equation in one unknown for you. Edit x and y are not variables they have particular values in your problem. But they are unknown until you calculate them

But, as elfmotat has pointed out that implies simultaneity, which depends upon the observer.

No offence but this is still rather contorted? Agreed, but what is the same time? Which makes the point, Mig and I were trying to put and explore the significance of this statement, given this is a thread about time. That's not actually what I said, but since you introduce world lines, do you consider it necessary to move the entire wordline in time for time travel to occur?

That's a bit harsh. You will find many so called "charge cloud" representations of molecules and atoms in monchrome in that classic textbook form Cambridge University Stranks et al : "Chemistry A Structural View" And a really beautiful colour one of uranium acetate on the front cover of Petrucci : "General Chemistry"

Surely that is prohibited by special relativity of simultaneity?

I suppose you could say it depends since 'chaos' is a collective term for a number of different effects. Some systems are constrained so that the chaos cannot 'grow'. Example of this would be the path of a metal ball hung above four magnets. In chemistry the Belousov-Zhabotinskii reaction. In other systems the 'chaos' can grow without limit even to the destruction of the system. Examples of this would be Euler instability of motion. Audio or video feedback. I agree with ajb that chaos and randomness are different phenomena. But also note that a small amount of initial randomness can lead to signification variation of system future history (called trajectory). There may, as ajb says, be entirely predictable equations that the trajectory follows. Any the chaos arises from (small) random variations in initial conditions. The course of this trajectory will depend upon the system as I said, but the instability that allows chaos is to grow or not, is inherent in the system, not the process. Euler instabilty is one such example. So called 'frequency doubling chaos', on the other hand contains the seeds of its own expansion in the process. Feedback comes to mind here. I expect this thread will develop further and I would be happy to expand on any of these points. I would be wary of using the term degree of chaos to measure it since fractals are often included in the basket of effects and the the term degree could be confused with the Hausordf Dimension, responsible for this phenomenon.

Note the formulae up to now have been about rotation about different axes. Your book may have been looking at volumes of revolution in different ways. Here is the same volume generated by rotation about the y axis in both cases. The area bounded by the curves x2=4y and y=2x is rotated about the y axis. Find the volume generated. solving the two equations simultaneously yields x = 0 and 8 ; y = 0 and 16 So the method on the left is as per my simple formula and the shaded area is rotated about the y axis [math]V = \int\limits_0^{16} {\pi \left( {x_1^2 - x_2^2} \right)} dy[/math] and for the second method we integrate with respect to x using vertical strips of area (y2-y1)dx which rotate about the y axis on a circle of of radius x . [math]dV = 2\pi x\left( {{y_2} - {y_1}} \right)dx[/math]

For volumes the corresponding formulae are [math]V = \int\limits_a^b {\pi {y^2}} dx[/math] for rotation about the x axis from x = a to x = b [math]V = \int\limits_c^d {\pi {x^2}} dy[/math] for rotation about the y axis from y = c to y = d

I was coming to that but I was waiting for you to answer my last comment because it is important. I just noticed that you said rotates about the y axis. For the formula I have given and you have used the rotation is about the x axis. For the formula elfmotat has given rotation is about the y axis, as he correctly states.

Are you sure your textbook says about the y axis?

You will also need to watch this when you come to multiple integrals, the limits become functions, not just simple numbers, and this catches many (including me when I first saw it)

When you change the variable of integration from dx to dy you also change the limits of integration. Since y = f(x) you can calculate the new limits, given their x values.

Yes that's correct, you are revolving the arc ab length Sab around the x axis to generate the area. It is simplest to understand if you start with a straight line parallel to the x axis (dy/dx=0) and generate a cylinder. You know the surface area is circumference times length of line ie [math]{A_{ab}} = 2\pi y\left( {{x_b} - {x_a}} \right)[/math] So you know the formula works. So all you are doing is having a fancy line or staking up hoops or circles to form the area.

OK do you understand where this formula comes from? ,

When I see you running through a brick wall or better a 38" Devon Cobb wall, I will believe it. BTW have you seen any of the 'Dynamo' productions? They are almost believable.

I am not sure if I have read your question correctly but If you do a simple definite intergration, between limits say [math]\int {ydx} [/math] This will get you the area beteen the curve and the x axis. It can be thought of as a summation of strips parallel to the y axis. similarly [math]\int {xdy} [/math] will get you the area between the curve and the y axis. This time you are summing strips parallel to the x axis. So you are calculating a different area. Is this what you mean by the variable of integration matters? When you are integrating (summing) disks to find a volume you can often arrange the strips to cover the required volume in either direction, so long as you pick the defining disk correctly. Have I pickup up your query correctly?

Right or wrong, the choice of negative for the electron is quite independent of the choice of direction of conventional current, which is another different convention.

This discussion is going somewhere. Let me throw something else into the mix. If you only have one spatial dimension you can get 'double booking' But we can count at least 3. So a physical object can sidestep around something on say the x axis by moving up the y axis, then past the object on the x axis, Then back down to the x axis.

Please don't try to pre-emp what I am going to say. It would be really helpful if you would indicate what continent your are on because the definition of Torque depends upon where and when you are. This is because American authors in the later part of the 20th century started using torque to mean what every one else had always called 'moment' (since before there was an America). So I will try to tailor my response to the information you might meet. It can be very confusing when different folks call something by different names.

OK so let us return to not just Flatland where there is (x,y,t), but Lineland, where there is only (x,t). Now on the above grids, studiot and michael 'occupy' the space between x=-1 and x+1. So my question as to where is the alternative observer amounts to How can an observer in motion pass the physical obstructions posed by studiot and Michael as she translates along the x axis? or Why does she not bump into the obstruction and stop?

Positive and negative are a convention or choice without doubt. But remember that we not only use positive and negative to denote the two possible polarites of charge, we also use them to denote directions in space or in the abstract that are totally independent of charge polarities. We have built up a complicated interlocking network of physical equations, based on the definitions we have today and all compatible with today's definitions and conventions. Many of these equations depend for their signs on more than just the assignation of + or - to the electron, but also on the combined effect of these chosen signs for direction. I am just exploring the implications of this.

If that was an answer to my question what am I supposed to think about it? The direction of conventional current is not defined by the signs assigned to the charge carriers. That is another convention which we are discussing. The direction of conventional current is a convention which we choose and then apply a - or + to (which reverses the direction or not) according to the sign of the carrier involved in equations involving this current.

So let us suppose that we switched the definition of negative and positve. What direction would you assign to conventional current, noting that actual charge carriers (whatever sign you assigned to them) would continue to flow in the same directions as before?

Would you clasify the outward normal as right handed of left handed?