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studiot last won the day on January 3

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About studiot

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    applications of physical sciences
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    Semi Retired Technical Consultant

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  1. In fairness to conjurer, Principia Mathematica is not enough of a reference by itself. Russell and Whitehead wrote the treatise with that precise title, following on from Newton, in around 1907. This was intended to be the pure mathematical encyclopedia of known mathematics at that time. Newton's work was actually about Physical Science and entitled Philosophae Naturalis Principia Mathematica Both are commonly referred to as Principia or Principia Mathematica. So before the crossed wires develop further let us all shake hands.
  2. Sorry, I realise that it was my mistake this time. I thought you were attributing my question to scuddy to scuddy, but I see now he repeated it.
  3. It's what Newton answered (to him, gravity was a force) Please sir, swansont sir, you've done it again. Didn't you like my poem? Here is a good account https://www.maths.ox.ac.uk/about-us/departmental-art/theory/differential-geometry
  4. No I think it is hiding the fact that there is no absolute time. Your problem is that you blythely say 'the clocks are synchronised', but do not define what this means. As I recall Einstein realised this difficulty and spent a couple of pages going through it in meticulous detail in his papers. He also realised that you can't have a time reference (absolute or otherwise) without a means to refer to it. Note I said absolute time not absolute time reference. It is these small apparently innocuous, things that make such a big difference.
  5. I don't know (or want to know) who the hell Bonner Corp are but the OP link takes me to a webpage called 'consent form' that requires me to accept unspecified uses by said Bonner Corp of my data, before I can proceed further. There is no evidence or promise of ever getting to the article.
  6. Consider a straight line. A straight line through the origin is linear. A straight line not through the origin is affine. This is because of the constant y = mx is linear but y =mx +c is affine
  7. I will need to find the references that led me to my comments before replying. That should make them more sensible.
  8. Surely this belong with your other thread on the subject? Anyway here are a couple of thoughts on the subject, extracted from a couple of books on the subject. Congratulations on being prepared to study books. Sadly a quality in short supply these days. Anyway first is from a famous texbook from lectures given at Cambridge University on Real Analysis. Secondly is from a modern text on Geometry, which has changed a great deal since Euclid and become largely algebraic. I post it because it is the bit about the axioms of vector spaces and shows a modern prsentation of what you ask in a vitally important subject, linear and affine maths. This goes someway towards your desire for a flowchart. It should be self evident which is which. So ask or discuss away after you have read them.
  9. The difference comes if the infinitesimal is multiplied by something suitable large in the final expression before it is discarded. Rigourous ? The dx that Wiki used is a 'dummy' that could just have easily been the 'symbol' I originally referrred to, for instance Ю. My point about the limit is that the limit of (A+B) = lim(A) +lim(B) exactly and [math]\mathop {\lim }\limits_{\delta x \to 0} \left( {2x} \right)[/math] is exactly 2x since it is independent of δx and [math]\mathop {\lim }\limits_{\delta x \to 0} \left( {\delta x} \right)[/math] is exactly zero So the result is exact with nothing discarded. I also realise that there is a correspondence (said to be) established between the reals and the hyperreals, in super advanced algebraic structure theory, which I hoped you would enlighten us about.
  10. I think it is truncated not rounded. I am having trouble with a conflict between the editor here and the Wiki extract I am trying to fix in my post. The paragraph following in Wiki clearly calls it a 'double number' and claims this to be rigourous.
  11. Going back to Wiki Note that it specifies "let dx be a non-zero infinitesimal" dx is not zero. However the rest of the procedure is just a smokescreen for the fact that in the last line the writer is going to ignore (drop) the dx. Talking about the 'standard' part is no more justifiable than choosing only to take the real part of a complex number. The alternative I was taught was to use limits and perform algebraic manipulations (including using the rules for manipulating limits) until the last line was reached [math]\mathop {\lim }\limits_{\delta x \to 0} \left( {2x + \delta x} \right)[/math] [math] = \mathop {\lim }\limits_{\delta x \to 0} \left( {2x} \right) + \mathop {\lim }\limits_{\delta x \to 0} \left( {\delta x} \right)[/math] Then and only then was the limit taken [math] = 2x + 0[/math] We developed these for quite a few derived functions. It can immediately be seen that the algebra is the same until the end but then nothing is discarded. It also shows why the principle of working out a few appropriate fundamental examples and then relying on developing an algebra of derivatives is the usual method taught.
  12. Actually I blame (at least partly) the current vogue to reduce the Maths to things like this [math]{G_{\alpha \beta }} = {T_{\alpha \beta }}[/math] (Baez on GR) [math]H\left( \psi \right) = E\left( \psi \right)[/math] (Hamiltonian formulation of Schroedinger equation) Those who have truly studied this stuff know that this is very shorthand hiding a multiplicity of equations and other stuff you would need to perform any actual calculations. But it leads others to think you can just write energy = whatever.
  13. Even professors of Physics have their off days. I am not a fan of JAK as a celebrity physicist, but he was trying to express the calculus of variations as applied to geodesics in three words. He could equally have said that the apple whilst attached to the tree is moving with the rest of the universe. Once detached it does not move at all. It is just that the rest of the universe moves past it the other way. A good question is not why does the apple fall ? but Why does it move at all ? Hint ask a poet not a physicist.
  14. It is indeed worth noting this as it demonstrates that the subject is not cut and dried. And there have been yet other versions of infinitesimals. Just as some have othere versions of infinitesimals, Thurston has other versions of extended numbers which he calls 'super-cauchy' numbers and 'neighbourhood of infinity numbers'.
  15. Although Hewitt may have coined the term hyperreal, he did no introduce infinitesimals. They go way further back, but here is an interesting treatment from Prefessor Franklin (MIT) in 1940.
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