studiot

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?

Just to clarify something for you. It is implied in what is being said about horizontal that posters are approximating the Earth (or Moon) to a flat surface in the region where the stone is being thrown sideways and that a horizontal line is a straight line at right angles to the vertical at the point of projection. Of course in reality the surface of the Earth etc is curved, not flat and a truly horizontal line is a curved line that follows this curve.

Actually I think there would be a difference as I said. Remember the OP only asked about the signs of positive and negative charge. This offset this change in our set of physics equations would mean changing the sign of the vector cross product. Such a change would preserve the electrical equations containing pseudovectors specific to electricity but not all cross product pseudovectors are to do with electricity. What about the outward normal to area or the torque pseudovector? Which way would they point if we did this?

A general balance of forces does not necessarily mean equilibrium. You need to complete your course in your Torque thread to find out more. If you must know, it is called Poinsot's Theorem.

A pretty good rendition from my Hymn sheet.

Where did this question come from, seems a bit odd to say that only one choice could not be the pH. I can see only one pH that matches all the indicators in the table, however, and it is not 7. What are your thoughts?

The eye-ring is a stop placed where the eyepiece forms an image of the objective. Its diameter is just sufficient to enclose all rays passing through the instrument. In a telescope the eye-piece comes just outside the eyepiece. The magnification or magnifying powere is the ratio of the objective diameter to the eye-ring diameter. The diameter of the eye-ring should not exceed that of the pupil of the eye.

This is a discussion forum and I made the point that physical objects are not points, they have physical sizes which show up as multiple coordinates referred to all axes. They have beginning and and end points and solid ones also occupy all points between. (I think it was Michael) made the point that he considers timelike and spacelike axes are not exactly the same and I was exploring the differences, in relation to my point above and also in relation to his comment about movement. You introduced simplified coordinate systems with (I think) one time and one space axis. Any such are continua and could be considered subcontinua of a space having a greater number of dimensions. (I think it was Jon) introduced Flatland, which is a larger subcontuum than yours, but still smaller than our 3 or 4 D world. I found this interesting and tried to incorporate this point in my discussion as well. Perhaps I have brought too many points together and we should consider them one at a time?

Classical EM theory is not quantised so any mention of photons is outside the scope. Discretisation of charge is admissible in classical theory since ions were included. The quantity called electric current' is carried by unspecified charge carriers and is considered as continuous stream of such carriers. This is equivalent to saying that the fluid elements are small enough to be considerd point masses for the purpose of fluid mechanics, so the mechanics of fluids is the mechanics of a continuum. Classical electrical theory including electric and magnetic fields and EM waves is also a branch of continuum physics. Waves are, of course, the bridge between QM and classical physics because they show mathematically how discretised phenonmena can arise naturally in a continuum. I don't think it is possible to discuss energy flows without some more sophisticated mathematics.

Forgive me, but I assumed you were restricting the discussion to a single (1+1) dimension contimuum by the above statement, not (1+1) embedded in a continuum of greater dimensions, which obviously contains many such restricted subcontinua. Please clarify. This is unfair. Of course it is relevent since it was (and still is) a legitimate question about the conditions of validity of your statement. Where is your second observer? In terms of your slant diagram the slanted lines still have a projection on the x axis, which I call their x coordinate. I agree that this projection will vary with the slant. If I zero a coordinate system at the left hand edge of my desk and align one end of my ruler with it, my ruler extends in space from x=0 to x=300, and occupies all points between continuously. If I now move it to the other end it extends from x = 1200 to x = 1500 and occupies all of the points between continuopusly and none of the original points. I am am inviting you to comment on the situation along the time axis, given that the ruler was manufactured in 2010 and will be destroyed in 2015. This is a thread about time so this is entirely relevent.

Well we will have to agree to differ on this aspect, since I believe in calling a spade a spade.

Good evening , MigL Yes and Yes But were your comments with reference to my post#59, that allowed 3/4 D continua (x,y,z,t) or with reference to mypost about Elfmotat's (x,t) continua?

Wouldn't there be a parity difference for pseudovectors like the magnetic field (pseudo)vector if we switched signs? I think this is what Sensei was getting at.