studiot

Yes indeed. But, with respect, that is because you are only looking at what you have said, not what I have said. Did you try my practical demonstration? No, nor have I said I disagree. I fully understand what you are saying but it is not relevant. You need to go beyond that to the logical consequences of the definitions and compact notation you are using. Things often look a bit different when worked out in full for particular cases. Mathematicians like to use compact generalizations, but Engineers have to make particular cases work. Yes this is the same idea. +0.1 for remembering it There are no virtual photons in Classical Physics and no the force exerted by a magnetic field is not mediated by photons. Paradoxically the force can be calculated by virtual work in the Classical Physics interpretation of a magnetic field, much more simply than a more advanced treatment, which yields the same end result.

You are just dodging the issue. Where is the scale axis the the scalar constants ak lie on, for it is not parallel to any of the existing axes xk. Try this practical demonstration. Take a piece of gridded paper, draw a pair of ordinary axes and mark the point 5,5. Now at the point 5,5 draw a vector of magnitude [math]\sqrt 2 [/math] at 45oso that it takes up zero room on the paper, because one thing is for certain That vector does not extend from x =5 to x = 6 and y = 5 to y = 6 You require a new coordinate system in a new space, joining the old coordinate system at one point only, to be able to accomplish this plot.

Back in the 1960s the Sunday Times published a chess problem every Sunday. Some of these could take (me) all day, and many were mate in two.

Thank you for pointing that out. +1 It shows how long since I last played, I even got the direction of play wrong. I had that pesky pawn going for queen.

Yeah that's about it alright except that the thing that is cataloged may have physical embodiment or may be just a catalog of numbers, like a velocity field. But yes we are very loose in mixing up the measurement of something with the thing itself in lots of circumstances. Edit Here are some thoughts on why we should not include the region of space in the field. 1) Consder the field of an electromagnet. If we turn the magnet off the field disappears, but the region does not. 2) What if we have more than one field in the same region of space eg a direction field and velocity field in a fluid? 3) What if we change the field values but not the region eg the temperature field in a bar changes if I heat one end. 4) What if I change the boundaries of a region of space? The values may alter or may remain the same depending upon the type of field eg both a fluid and electrostatic field will expand to fill expanded boundaries. The fluid values will, in general change but the electrostatic ones will not.

I don't know what is going on here, but this is the third time I have posted this. 1) Rook to e3 check 2) If bishop (or knight) to e4 then 3) Queen to f6 mate.

The last time I tried to quote and delete unwanted sections of a post to make the replies relevant it broke the forum programming. So apologies for writing my replies into your post.

Yes I know but you wanted one and the rule I remember was in exactly 2 not in at most 2. I will work on it again when I have finished responding to other questions. OK I now make the first move rook to e3 - check. Blocking with the bishop is countered by queen to e6 edited

Sure, I originally said that a field 'exists' in a region of space, that you later called a manifold. So we need a coordinate system to identify each point in the region or manifold. The field function then assigns a particular value to each point we identify for the field variable. In Physics that field variable may need only a numerical scale or it may need a whole coordinate system of its own. So a temperature field assigns a number temperature on a temperature scale to each point and this scalar field thus has a one dimensional field variable. A more complicated field variable, say a fluid flow vector, will require another separate coordinate system to be established at every point in the region, in order to be able to define the vector. There may be a transformation to allow us to move between the coordinate systems. This will happen if both are of suitable physical quantities. If this is the case the transformation is called a chart. There is a chart for each point in the original coordinate system and the entire portfolio of charts is called an atlas. There will also then be a relationship between each pair of charts in atlas. This transformation is called a Connection. Of course it may also be that there is no transformation so it is impossible to turn temperatures into spatial positions, and vice versa. Glad you are finally happy. I secretly hoped you would read it.

Aha. I see I've set my pegboard out with the castles switched. That's what makes it fun.

Yes it looks like it the first move should have been b7 to c8. I thought I was too quick.

Is the bishop at g5 sleeping?

So queen takes pawn?

Seems to work OK to me in two moves. The black king can't move on his first move and the black bishop prevents mate in 1 so removing it seems a good option. But thank you for the references.

To help you complete your research into this question and work towards a better answer, here is some history. De Broglie's hypothesis (1924) was originally proposed to explain the photoelectric effect. Now the proposal followed Planck using the equation E = hf where f is the frequency of the light concerned and h is Planck's constant. Of course there is no relative speed issue with light. It has a constant speed c for all observers and sources. So using the equation c = fL where L is the wavelength, it was possible to arrive at an expression for the momentum of a photon. Relativity gave one expression for the 'apparent mass' of the photon and the momentum another, equating these eliminates division by zero mass and produces a relationship between p = hf/c = H/L = Hk where p is the momentum, H the reduced Planck's constant and k the reciprocal of wavelength or wavenumber. De Broglie further proposed that this be extended to all particles ie including massive ones and the same calculation be done, but this time including not cancelling the mass to yield a wavelength for such particles equal to h/mv. Now the velocity input to these particles will be relative to the medium the particle is travelling in. So for electrons orbiting the nucleus in its electrostatic field this will be the tangential velocity associated with that rotation. For particles passing through a cloud chamber or other matter it will be relative to that matter and so on. There there is no problem with the 'relativity' of the motion. Incidentally these associated 'matter waves' were experimentally confirmed by Davisson and Germer (1927) and Thompson (1928)

I would still not rule out river flood transport. Near where I live, there is an ancient clapper bridge across the river Barle on Exmoor. Winter floods have shifted/transported large stones weighing several tonnes, downstream from the structure again. The river Barle is an inconsequential trickle compared to your american rivers in question. http://www.bbc.co.uk/news/uk-england-somerset-38844162

I used to enjoy chess problems, they are hard to find these days but this one was too quick so I probably missed something. But isn't the first move B7 x H1 (Bishop takes Bishop) ? +1

Sometimes its good to bring some simple sanity to a thread. +1

I had always understood that Roemer was the first to measure this. https://en.wikipedia.org/wiki/R%C3%B8mer's_determination_of_the_speed_of_light

Thank you Mordred for that large list of net references. Since I normally go with books, here are a couple to complement them. Firstly The Standard classic for the heavier mathematics you refer to from the founder of this sort of theory, O,D, Kellogg Foundations of Potential Theory O D Kellogg Kellogg not only develops the mathematical theory necessary for those fields that are amenable to Potential Theory (ie have Scalar, Vector or Tensor Field Potentials) but also offers much insight into the connections with some of the more esoteric pure maths behind it such as The Heine Borel Theorem. It is strictly an analytical approach in the mathematical analysis sense. Secondly this delightful tome from the Engineering sector for those using the techniques rather than developing them Engineering Field Theory A J Baden Fuller Probably more appropriate here, this book offers not only practical insights into using the theory and solving the equations both analytically and numerically since engineers need to use equations rather than study their pedigree. It covers a wide range if Fields used in practice and includes a large table of the relevant formulae from all the different chapters about different sorts of Fields. I offer the following brief extracts that may be of help here. First the chapter list Then two pages from the introduction that all here can read and hopefully draw something from

What exactly is a kinetic energy field? You have a potential energy field because a massive ( massive = possessing mass) body exerts a gravitational effect on every point in the region around it. There is something called 'complementary energy' in continuum mechanics but it is not really a mirror image.

Since you ask, try The Boundary Element Method Brebbia. Yes, the one defines the other. The point is that it can be easier to perform the calculations over a few boundary elements than many internal mesh points. Integration over a closed surface or loop is often computationally easier than a volume integral. Do you need references for this stuff? The angular momentum remains constant but the components change. That is as the radius decreases the rotational speed increases. This is not an energy phenomenon. Moving a particle from point A to B removes potential energy from the particle or changes it to kinetic energy. A good example would be the drop hammer on a piledriver. The gravitational energy lost by the hammer in dropping is transferred to the pile. This is replaced when the cycle is completed by raising the hammer back to drop height.

I was not going to, but, Xerxes, since you bring it up A field in Physics can entail two quite distinct and different coordinate systems and usually does. No transformation exists between these coordinate systems. I do think such discussion is way beyond the basics suitable for this thread, but I will elaborate if you wish.

Thanks, Strange, for explaining what this thread was all about. +1 I nearly missed it, so word to the OP - using a more descriptive title will encourage folks to read (and contribute to) your thread.

Like I said, I expect a block of posts was moved. I'm not even sure how far they can sort individual posts within a block. A separation line has to be drawn somewhere and sometimes things get tangled up in this. I have 'suffered' this myself so it is not personal, However I would like to say that I, and quite a few others I think, value you contributions here.