studiot

Why can you not simply take the average, rather than mess about with moduli? If P is halfway between R and Q then [math]{X_P} = \frac{1}{2}\left( {{X_R} + {X_Q}} \right)[/math] and the same for YP So substituting your values and doing a bit of manipulation [math] = \frac{1}{2}\left( {\frac{{a\cos t}}{{1 - \sin t}} + \frac{{a\cos t}}{{1 + \sin t}}} \right)[/math] [math] = \frac{1}{2}\left( {\frac{{a\cos t + a\cos t\sin t + a\cos t - a\cos t\sin t}}{{\left( {1 + \sin t} \right)\left( {1 - \sin t} \right)}}} \right)[/math] [math] = \frac{1}{2}\frac{{2a\cos t}}{{1 - {{\sin }^2}t}}[/math] [math] = \frac{{a\cos t}}{{{{\cos }^2}t}}[/math] [math] = \frac{a}{{\cos t}}[/math] as required I will leave you to show the same for the Y values.

I think you should focus on the bigger picture. You are asked to sketch the curve. It is therefore reasonable to assume that you have been taught some methods and procedures for doing this. You should tell us what you have achieved in this respect. Definition and use of the term 'domain' are not introduced until university in some countries, although the idea is very simple. The domain is that part of the x axis where the equation makes can be evaluated to provide a y value. So follow your procedures for curve sketching and it will lead you to the rest of the answer for the first part.

Ripples are surface waves, and waves are a means of transferring energy. The shape and speed of the wave are controlled not by the initial disturbance but by the properties of the medium, in this case the water. The amplitude (strength) of the waves are controlled by the energy of the initial disturbance. This disturbance of course is the fact that the stone has kinetic energy, most of which it loses when it hits the water and slows right down. It is this energy which travels outwards as expanding ripples.

Thank you for this information, MigL, I was not aware of this work. Isn't there a lot of work going on in many directions? However I don't see that a flat torus provides a manifold that does not require a higher dimension to exist in. This is certainly acknowledged in the analyses I have seen, eg http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html Such an object is only locally flat, in the sense that a straight line is an arc of a circle of infinite radius. You need to find a globally flat universe, otherwise the triangulation exercise I described will still show discrepencies. All 3D objects possess 2D surfaces, you cannot have the surface without also having the object. However there is a more promising approach as a candidate for Strange's manifold requirement, which is essential a desire for the surface without the object. If you choose a space filling curve you have a manifold but no object, although the 'curve' occupies 2D or 3D space. The curve can be considered an ordered list of points in the space. http://www.dcs.bbk.ac.uk/~jkl/BNCOD2000/slides.html An interesting consequence of this is that 2 points can be adjacent in 2D or 3D space, but not be neighbouring points in the manifold. (Plenty of scope for scifi here) go well

You need to formulate your question more exactly. "A particle is moving.." With steady velocity? Accelerating? What? "If the helix is expanded" How? Radially? Axially? Is the spring subject to the radial contraction that would accompany being pulled out along an axial direction? Is the particle subject to the same forces that expand the spring or how does it receive its change of direction? Why is it moving, is it like a bead sliding down a vertical spring or what?

It's not a question of belief, that is for religion. it's a question of observation, followed by mathematical deduction. (not the other way round as that can lead to error). Of course the surface of a finite sphere is finite and there are no bounds in the surface. So what? Further this is not true of any folded plate structure like your pringle. A being on that surface could very easily take measurements on sets of three points and deduce that the distances between them do not match a planar triangle and that the area of the figure they enclose is subject to spherical excess. This would lead to the deduction that there must be a third dimension allowing curvature for this to happen. Yet that same being could develop a consistent system of mathematics on his surface. But he could not claim it to be a complete system of mathematics. Consider the following. The rotations of a plane triangle form a group and allow a consistent algebra. However for the triangle to rotate, there must be at least a disk of diameter equal to length of the side of one triangle available.

Well the original version ran along the lines of Mick the Martian says all Martians are liars. Which is a combination of the statements. Mick is a Martian. All Martians are liars., which may posses independent truth values.

Do you mean computer programs like LEAP and STAAD ?

As with car bodies you don't need to remove the rust, just passivate. That means remove the loose mechanically. Then chemically treat the rest which is firmly bonded. This changes the oxide from the brown variety which is flaky and non protective to the black variety which is safe. Of course you may be able to install cathodic protection. http://www.epa.gov/oust/ustsystm/cathodic.htm

It can actually be much much worse than that. Ferrous metals are subject to so called pitting and pinhole corrosion where the rusting bores straight through the sheet metal forming a fine hole. You see this on car bodies. Obviously a series of holes will compromise your evaporators long before the main sheet is rusted away. You need to mechanically work down into pits as the active part is right at the bottom of the pit. If possible treat pits with phosphoric acid to passivate and wash out. You should consider the possibility of a corrosion inhibitor if at all possible

Taking the geometric equivalent of saying we can have a full (possibly infinite) set of numbers without using negative numbers. This is true, but we cannot claim we include all the numbers or all their properties. Using intrinsic geometry is the geometric equivalent. But you cannot then declare your subset to be the universal set.

That's a nice pic, ccweb. It shows all four laws in action at once, though you have a slight misapprehension about what they refer to. Newton's laws apply throughout and can be used to derive Bernoulli's equation (though there are other ways). Bernoulli's equation is a direct consequence of Newton's laws. Newton's laws, however, are not directly responsible for what you call the downwash (which is not a force but a movement of air). That is due to the clockwise Jukowski circulation around the airfoil. This is important because it explains the velocity difference. The circulation flow is up in front of the airfoil. left to right across the top, down at the back and right to left underneath. As a result the circulation flow assists the air stream over the top and opposes it underneath. So the velocity is higher over the top and lower underneath. So to the fourth theorem, that of Kelvin. To the right behind the airfoil you can see the wake of trailing countervortices. They are revolving in an anticlockwise direction. This reduces the overall rotation or vorticity back to zero along the streamlines, in accordance with Kelvin's theorem. And talking of streamlines, MigL, Bernouilli applies along any streamline or streamtube, not just those in pipes. But remember it applies only to parcels of fluid along the same streamline, but at different locations or times and only in steady flow. The pic also shows how the positive angle of attack leads to a clockwise rotation for the circulation. This is because the first part of the airfoil to encounter the flow is the leading edge. This solid object wants to deflect the flow up and down but comparing the two paths it can be seen that above the airfoil the air is clear, but below the deflected air is 'obstructed' by air gathered under the airfoil by the next part to encounter the flow. This obstruction = higher pressure. I am sorry that last was not well explained, I have been meaning to draw a series of sketches showing how the circulation develops as a consequence of the attack and this gathering process.

Is recursion a valid method of determining truth values? How about the following program line: 100 If you have not reached this line end program. Incidentally the liar paradox contains 2 statements rolled into 1 so you have the difficulty Determine the truth value of (2+2=4 and 2+2=3) Clearly unresolvable, but quite resolvable if split into component statements.

I was just starting to add to my post when the phone rang. Yes you can restrict the properties to create an 'intrinsic geometry' see for instance O'Neill @Elementary Differential Geometry' p271, Intrinsic Geometry of Surfaces in E3. However we have that word 'restrict' and we are back to the difficulty of 'everything'. Take the surface of a sphere. It is (in some ways) isomorphic to the plane. But It was also known to surveyors by the mid 18 century that a line of constant bearing or heading was not the shortest distance between two points on the Earth, as it would be with the plane. In fact it is an S shaped curve on the surface of the Earth called a geodesic. In this way some properties of the embedding higher dimension are required.

Go on do tell.

You need to complete your 2D analogy, not pick part of it. You can only have the surface of a sphere as a 2D surface if you embed it in 3D space. The boundary is then between the inside and outside of the sphere. So all points are boundary points in 3D space, and of course you must have 3D space. Again this is not a problem unless you are claiming 'everything'. This must then include the rest of 3D space. In our 3D world manifold, we need to introduce 4D space to accomodate this. And the same boundary issue immediately arises. A further complication with this view is that it is possible for a 2D 'being' to detect the effect (existence) of 3D space by such physical phenomena as shadows. But we have never observed these in our universe.

I don't disagree with any of this, but would like to add some extra material. Not all aircraft can fly inverted especially helicopters. This is despite the recent movie from Hollywood. Those that can are not as nimble (could any actually take off or land if they had the gear?). As noted above they have to adopt an odd attitude to counter the inverted tailplane contribution and any assymetries in the main airfoils. It is good to see that someone is observing that there is more than one physical law obeyed. That is it is not Newton v Bernoulli, since both are obeyed. Of course Kelvin and Jukowski are also obeyed. As regards the pressure and velocity, there is not a single upper and lower pressure and velocity. Both are subject to a large variation along the airfoil. Further not only is there a relative difference, but in general the air above is less than average stream pressure and the air below greater than average stream pressure. The attached sketch shows typical detail (the airfoil has a valid angle of attack and will not be horizontal)

Thank you both I am aware of this and pointed it out in post#5 and repeated it several times since in this thread. However you miss my point. 'Finite but unbounded' is a glib phrase that applies to some aspects of the ball but not all of them. In particular the longest line you can draw on the manifold is finite and bounded (by its length). If the manifold expands then one of my conditions for expansion must be met. ie can the line get longer? You need to consider carefully the implications of a boundary.It is not a question of Ricci tensors or high falutin maths, it is a question of some basic set theory. The temperature scale is comfortably within the mathematical framework because with the temperature scale we are not claiming 'universal properties'. We can establish a temperature scale that is bounded below by absolute zero, but choose an open set of temperature values that do not include the boundary, without penalty. However. Set theory, for continuous sets such as we are discussing here, identifies two types of members, interior members (or points) and boundary members (or points) Interior points All interior members only possess other set members as neighbours or all the neighbours of interior points are in the set. Boundary points Boundary points possess some neighbours that are memebrs of the set and some that are not members. That is not all the neighbouring points of a boundary point are in the set. Thus it is no problem that -273.15 is not in the set but -273.12 is in the set of all possible temperatures, where absolute zero at -273.14 is a boundary point, -273.12 is an interior point and -273.15 is not in the set. However it is a real problem that if N metres is the longest line that can be drawn that a line of length (N+1) metres is not in the set of everything.

Not at all. Read again my last post and post #5. For expansion to occur Either there are more metres or each metre is 'longer' or there is some other agent also at work as with the temperature scale.

Of course it makes a big difference. Consider my first question, which might be used to stand in for 'the diameter of the universe'. Case1) The universe is infinite. Then there is no such N as I described. The number of metres is unbounded above. And what of the mathematics of such an 'N' ? It means that whatever value of N is proposed I can find a larger N+1 within the existing system. What then is the meaning of expansion or making N larger? ie what is the 'value' of infinity plus one? Mathematically it is still infinity. Infinity is capable of sustaining an infinite level of expansion, without appearing any different. Case2) The universe is finite, is more interesting. It automatically brings in the unavoidable issue of the boundary, since then N is bounded above and the question of what about an M > N arises. Mathematically this question can be avoided by considering an open set, ie excluding the boundary, as in the frst example in my post#5. Alternatively you can address this issue by fiddling the metric as in the Poincare disk model (which also excludes the boundary). However if you exclude something, how can you describe the universe as including everything?

Strange, once again I thank you for putting all that typing effort in replying to my questions. Unfortunately you replied to neither, since you answered the the second as I don't know However I am quite certain that it makes a very big difference to the mathematics whether the universe is finite or infinite. Further since you do not know the answer to this you cannot state that the mathematics is A or B since finite and transfinite mathematics is different at the most fundamental level. In particular you cannot answer my first question until the second one is answered.

Thank you for the link. However I saw many words and little maths, really only a few unsubstantiated formulae. There was no development of a mathematical argument. Are you suggesting I should learn this? Certainly there was no mathematical answer to the straightforward mathematical question I raised, although the article did state that there was no expansion (section 2.0) Unfortunately the article introduces an aether theory in section 2.1 et seq, which substatially diminishes it in my view. I would add that my question is linked to my comments in post#5 and represents the beginning of a critical examination of the idea of expansion. I sympathise with the OP's dilemma and seek some solid mathematics.

That's all very well but that does not explain what is meant by 'expansion'. Do you consider the universe finite or infinite? By that I mean is there at least one longest line, of length N metres (however large) that can be traversed in a constant direction without passing the same point more than once.

Perhaps you don't know what Bernoulli's equation states? Another common misapplication of Bernoulli's theorem is to attempt to apply it to two unconnected places within a flow. Bernoulli's theorem is about energy conservation along a streamline. The streamlines above the object are not the same ones as those below. So you cannot just take values from the streamlines above the object and substitute into the streamlines below.

I'm sorry, where does Bernoulli suggest rotation? He doesn't, period. Jukowski's Thereom 1906 does Bernoulli's equation is a scalar equation. The Lift equation is a vector equation. Jukowski's theorem and the Circulation theorem are both vector equations. You cannot use a scalar equation as a vector one.