studiot

Well Xerxes has certainly spelled it out for you in gory detail (What he said is true), But one word of warning. Xerxes has not been lazy, he has been kind to you and written out all the summation signs. (he has actually put a lot of work in) Tensor addicts have a secret convention that they do not bother with the giant sigma sign, they regard it as 'understood' whenever you see the double suffix. Let me know if you need a translation to rough guide English.

I think the original proposal has now been disproved by many good counterexamples. However the subject of pattern, symmetry and symmetry-breaking is interesting and worthy of more discussion. Here are a couple of extracts from Philip Ball The Self-Made Tapestry, Pattern formation in nature. The interesting idea is that patterns form as a result of symmetry breaking or reduction of symmetry and that the most symmetrical patterns are the most boring. He gives many examples. So the circle has the greatest symmetry of any plane figure, a snowflake is the result in the reduction of symmetry. A uniform gas is 'randomly symmetrical'; apply a symmetry breaking force and the structure becomes more complex and more interesting.

Nice counterexamples +1

Be careful what you wish for, lest it comes true. H+ is only a bookkeeping quantity in relation to water. Look here https://en.wikipedia.org/wiki/Hydronium (also calleed hydroxonium and here https://en.wikipedia.org/wiki/Hydrogen_ion Safety note Passage of electric current will only get you hydrogen and oxygen gas and a large bang if you are not careful. Specify your needs more thoroughly.

I am not an expert in astro work, although I have carried out both gravimetric and astro surveys. Thus I asked a real expert. Are you such an expert ? If not, why did you presume to lecture me as though you were? It is fairly obvious to anyone who has done gravimetric work why swinging pendulum clocks are fine for measuring little g, but inappropriate for measuring big G. It is also obvious to anyone who has ever set up or moved a grandfather clock. If you don't know the answer to this, why don't you simply ask? I'm sure Strange is bursting to tell you.

For those that like reading, Roger Penrose devotes a whole chapter (chapter 29) of his book The Road to Reality to this question. He examines 6 different interpretations, including Copenhagen, in some (mathematical) detail.

The following might help. A space is the set of all possible values, whether we want them or not, of a given condition. So the usual cartesian 3 dimensional space is the set {x,y,z} where x, y and z take on every possible numerical value. x,y and z are then said to form a basis for the whole space since we can generate the entire catalog of triples from them. We can restrict this in two ways. We can select a subspace of the whole space. For instance the plane z=0 is a 2 dimensional subspace of {x,y,z} since it ranges through every possible value of x and y and does not need or use any values of z. This subspace is also a subset of {x,y,z}, but not all subsets are subspaces. The cube bounded by the six planes x=0, x=1, y=0,y=1, z=0, z=1 is a set of triples, {x,y,z}, where 0<x<1 , 0<y<1 and 0<z<1 Naturally there is a difference in some rules for subspaces and subsets or there would be no point in making the distinction The difference between a subset and a subspace is important in the definition of real world fields, which can occupy a subset or subspace.

Also make offline backup copies of all your datafiles now in case the virus also introduced one of those ransomwares that become active after a month.

How is your understanding of matrices? I think they are a good place to start for those who want the Physics, but not the detailed maths. Most tensors in the physical world are second order so can be written as matrices.

If you haven't already done so, delete all non Microsoft browsers, especially Chrome, restart and then reinstall after the full cleanup. Some further tips. Revo Uninstaller is better than Microsoft uninstaller If possible run CCleaner as soon as you have enough control of an infected machine. This substantially reduces the amount of crap malware scanners have to wade through, speeding them up. Run it again last thing after everything else. Run HitmanPro as well as adw cleaner. You will have to manually delete anything it finds or pay for the autodelete. The french program JRT.exe is also a good rogue finde in the early stages to help regain control of an infected machine.

No, just as examples the stress tensor, the strain tensor, the dielectric tensor, and the inertia tensor all have 9 elements and the same form but very different units. Tensors properly don't have units, but their elements may have units. Some may refer to the units of the elements as the units of the tensor. Some tensor elements are simply coefficients or just plain old numbers, some have units. I think all the elements in a particular tensor must have the same units as each other. Perhaps someone else will confirm that. Having the same units as each other does not necessarily make the elements of the same type, for instance the stress tensor contains shear and direct stresses which are different, although they enjoy the same units. Well sort of. Tensors are essentially point functions so some tensors do not involve a cell at all. Those that do involve a differential cell in the calculus sense that shrinks to a point in some limiting process. Some 'cells' are composed of differential elements, dx, dy and dz in the engineering 'control volume' sense and actually exist in the same space as the x,y and z axis. That is they have measurable length along these axes of dx, dy and dz. For some quantities resort has to be made to the phase space referred to in the post#88 in the Fields thread.

To move this thread forward let note that it is a spin-off from a history thread. So to look back at the history of gravitational measurements I note that from Kepler and Newton to the 1930s the main thrust was to determine big G, the universal constant. Here is a table from Newman and Searle (1957 edition) : The General Properties of Matter : showing the state of knowledge at the end of that lengthy period. Between about 1880 and 1930 another g, little g, was coming to prominence as the science of Geodesy matured. It can be seen from the table that D, the average density of the Earth was also becoming considered when the last (Heyl) measurement was listed and that now new information was added on G in the next nearly 30 years. Newman and Searle also give an account of the detailed study of the factors that might be able to affect the value of G, in the latter part of that period. Good quality gravity measurements are not cheap undertakings so I surmise funding considerations played a part Geophysical exploration by gravimetric surveying was becoming possible so the impetus moved from big G to little g. For instance Clark at the NPL obtained an accuracy of 2 parts in 10-8 in 1940. The measurement of little g does not suffer the same objections to I am not an astronomer so I will leave it to a real one to tell us what is accurate enough for astro/cosmological purposes.

Although you will win no prizes for fine prose, I see no reason for anyone to mark this down twice. Particularly as it contains the good manners to thank someone (me). So +1 from me.

Not intending any condescension. I am trying to pitch my replies at an appropriate level and I seem to have hit that mark with the extract from Baden-Fuller. Exchange or mediating particles form no part of classical theory, so I don't think your question was stupid at all, just off topic. Since I have not managed to explain my contention that the field variable may require a space of additional dimensions I will try the extract method again since I have found someone else who may explain it better. Here is a short extract from Beiser : Concepts of Modern Physics. The author has the knack of introducing just enough Mathematics at just the right level to promote understanding without obscuring the Physics. Not only that but the excerpt provides a bridge from classical to quantum theory that you seem to want. Note he describes the situation for a system of particles, using position and momentum. You should replace that by a system of values of the field variable, using appropriate phase variables for the particular field. Note that this can incorporate the derivatives Xerxes referred to - I never denied this only meant that it is not the whole story. Hopefully readers will get on better with this.

All Physics theories are models and suitable for some purposes but not for others. This subforum is about Classical Physics and the models are not suitable for QFT. If you want to study Quantum Field Theory there is a sub forum provided. You should be aware that models may differ in the answers/solutions they offer to particular questions and we chose the most appropriate by experimental comparison with reality. Sometimes the models offer the same answer but one is much more computationally difficult than the other. Would this one not then be the one of choice? Here is an example: An electromagnet lifts a car to the crusher. If we model the electromagnet as a slug of ferrous metal encased in a solenoid and the car as a free slug of ferrous metal, provide a formula for the necessary current to lift the car and the resulting magnetic field between the electromagnet and the car. Now this takes a few lines using Classical Theory and I once had a debate (elsewhere) with a Doctorate in this subject who visits here occasionally. He said no magnetism does not work like that and produced at least three pages of fancy more advanced mathematics to come up with the same formula! So if you want to learn more about the subject, all well and good. There are several lifetimes worth of material out there. But please don't try to pick off odd isolated statements to try to prove your (mis)conceptions about something.

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.