studiot

In what way

So how do you address my comment ?

Can you solve this equation x2 = 9 I hope so because my next question is what is x2 ? My answer here is that is it a function of x f(x) = x2. There are lots of possible equations involving functions of x. A function is an expression or formula that tells us to do something with x. In this case to square it. Some are useles, boring or otherwise uninteresting. Derivatives are functions of x. So we can write equations involving the function that tells us to differentiate x. A very simple one is dy/dx = 5. Solving it tells us we have a straight line with a slope of 5

Suppose two people started at a point on the earth's surface: the first travelled 100 miles north then turned and travelled 100 miles east. the second travelled 100 miles east then turned and travelled 100 miles north. Would they be in the same place at the end ?

Yes I expect there are a fe people somewhere that consider this to be so. But in myopinion they would be wrong. Whatever else our universe might be of possess it is not empty. As soon as there is anything in a universe that anything must have a configuration and that configuration must possess energy of configuration, otherwise known as potential energy.

Wasted energy ? Is your consideration in terms of wated money or wasted Kw-hours ? On the subject of energy waste there are surely bigger considerations ? Are you cooking by gas or electricity or some sort of liquid or solid fire ? My current costs are £0.10 per Kw-hr for gas and £0.31 per Kw hr for electricity.

Yes you are considering the right things this time. As regards the start point, even simple counting has a start point 1,2,3........ But it has no end point unless you run out of numbers to count., which of course you will not. That is what is meant by infinity in this case. (remember there are other cases) An absolutely spot on answer to the series part of the question. Clear compact and brings out the essential points. +1 I would add to this that for further information @Boltzmannbrain should study Cauchy sequences as this is one good way of dealing with this subject. https://en.wikipedia.org/wiki/Cauchy_sequence

I meant to say +1 to Markus for his addition before. Mathematically ? see here since it was Cantor who I think first defined manifold. BYW I'm still not sure if you mean find out the dimension by observation in which case I would recommentd looking for shadows or considering the commutativity of rotations or by theory in which case I would refer you to the topological idea of shrinking a fundamental area to a limit as described in Needham's book we have been discussing.

I'm so glad you guys had this conversation whilst I was considering my answer. Like swansont, I originally thought you meant physical space and how you would measure the dimensions by physical observation. But 'space' in Mathematics is different from 'space' in Physics. Whatever the space you are referring to you would need to have more detail to proceed. In pure mathematics, space refers to a master or container set for several sets which make up the space. These are not subsets, since the nature of the elements in each is different. You need at least set of elements or points, a set of axioms and a set of relations between the elements. This gives rise to different mathematical spaces eg geometric space, hausdorf space, phase space and so on. These rules will enable the mathematical determination of both the meaning and number of dimensions of that space.

I'm not clear on exactly what you want to do. What do you mean by 'electrically conductive length' ? If, as you say, A and B are at fixed locations, and you wish to provide a current path from A to B a curve is the only way to provide variation. Are you aware of the difference between rheostats, potentiometers and variable resistors and their connections ?

Difficult because Gauss require a closed n dimension 'shell'. You can describe a closed sperical shell, which is 2D and curves in 3D. But it is not possible to surround an object in 4d with a corresponding 3D 'shell'. There is always a route through the 4th D from inside to outside. Flux refers to contours or imaginary equipollent lines in the normal way connecting points in the manifold.

last lines Gnoses mare Thebe trux

Is the constant factor mean over time? ⍙Y is just the change of the y axis over time Time doesn't enter into it. I foot is identical to 12 inches. If you prefer 100 cents makes one dollar, period.

One of the consequences of Gauss is that the view from outside a shell is independent of the disposition of the flux charge within the shell. Newton is not required for this and it also applies to fluxes other than gravity. In turn this leads to multiple, if not infinite, solutions. I see it as bit like the different between the moment of inertia and product of inertia. There are an infinity of solutions to question what disposition of mass yields a given moment of inertia, but only one to the question what yield a given product of inertia. Somewhere there is more to the story.

Wouldn't this contradict Gauss' shell-flux theorem ?

First day at school My kettle just died RIP boiling water You will be mist

This is where some very basic science comes in. You do not need college to know that there are many forms of energy and that all energy is measured in the same units. So the conversion of one form of energy to another is a quantity of energy, which is still measure in the same units. yes I know that over the centuries there have been many different measurings units for energy, just as with say length measured in feet, inches metres, links, chains furlongs and many more. But each can be converted into the other by multiplying by a constant factor, as the number of inches is 12 times the number of feet. Does this help?

emergent from what ? As I understand 'emergence' it arises because things have properties and emergence arises in very special particular cases of combinations of these properties.

Which is why you should put in the effort to study some very basic science. It took thousands of Men thousands of years to develop the knowledge and understanding of Physics that we have today. How long do you think it would take one man on his/her own ? Guessing is a very very ineficient way.

Very Euro-centric of you. The middle east and the far east had no such dark age. Overall, world-wide, there was no regression of technology. In fact, gun powder was invented during that period (9th century). Really ? I suggest you check your arithmetic. Let us say there was a small amount of progress in the largest continent, Asia. But this is only 31 million square kilometres. Set this against 29 for Africa, 21 for North America, 17 for South America, and 8 for Oceania (ignoring Antartica which has not been settled in human history) that makes 75 million square kilometers where no advances were being made.

If you would like to Stop, take a deep breath, and count to 10. Then try to reformulate your query so others can make sense of it it would be very helpful. Why do you think total mechanical energy would change over time ? The law of conservation of energy is actually more fully the law of conservation of mechanical energy of an isolated system and it states that the total mechanical energy of such a system does not chnage over time. Yet you have not stated what system and circumstances you are applying your thoughts to.

Good points, Ken. +1 But I think you should go back a little further than you suggest for the history. The groundwork was laid in the 1600s by Hooke and Newton and continued in the 1700s by the Europeans, starting with Farenheit. This lead onto Kelvin's Absolute temperature in the early 1800s. So by the time of the French- Spanish geodetic expedition to Peru temperature measurement was routine. They made the first systematically recorded temperature observations in South America. Hooke produced the first standard rain guages, wind guages, temperature gauges etc necessary for properly established scientific recording. Here is his scheme.

Yes indeed at one time there were lots of thermometers around. They were unevenly distributed over the globe, depending upon the resources of their nations in charge of their location. Also the readings were of uneven quality and some readings were more frequent than others. Readings were also taken at sea and more recently in the air by aircraft and ballons, manned and unmanned. Today we have the benefit of continuous near blanket monitoring by weather satellites using remote thermal imaging techniques. Both the average and local temperatures are known to affect various marker such as plant growth and pollen production, ice thicknesses and so on. Records of these from ice cores and fossils are used by paleoclimatologists to establish past temperatures. Does this help ?

OK so before I give any detail a couple of important things you need to know. There are two editions of VCA, which was first published in 1997. VDG&F followed in 2021 and was written in the same style and from the same viewpoint. A second edition of VCA (which I have) was published in 2023 and has some important typographic deficiencies remedied. The maths text is the same, but the many diagrams now all have explanatory captions, the index has been much expanded and a conventional referencing system added, and the book is in a larger physical size. The two books contain much common material and frequently reference each other. Between them they develop the author's theme that it is good for understanding to approach fundamental principles in maths from multiple viewpoints. He expresses how reassuring it is to come to the same result from different routes. Both books have Feynman's american ability to pull out the essential statements in a clear and obvious form and to highlight and separate them from the block of the text. I would certainly recommend both as a pair; you really need the 2023 version of VCA though. I already have three books entitled "The Geometry of Complex Numbers", but apart from the usual few diagrams you might find in any work on complex analysis they do not approach it from the geometric viewpoint at all. VCA certainly achieves this. VCA covers a lot of ground with good mathematical insights. But what it is not is a tabulation or treatise on the applications of complex analysis, which is where many readers are coming from. So if you want CA in the solution of differential equations, it only mentions 2, Schrodinger and Dirac, complex Bessel functions are not treated at all, you will have to go to alfhors for that. Complex integration is dealt with at a fundamental level, in relation to measure theory. It is not a textbook of complex integration techniques. (Conformal) mapping again much wanted by engineers is treated at a mathematically fundamental level rather that a catalogue of techniques. As promised the book is not a catalogue of algebraic results and formulae, with the excuse 'we can explain this result geometrically'. The results are there but arise naturally by considering the geometry ( my preferred way ) and also the topology / continuity. Also arising from this geometric approach Tristran delves deeply into noneuclidian geometry. Hope this helps, sorry it has taken so long to reply, but I needed to do justice to the books.

Thank you for this reply and the other posts you have recently made. I hope I am now clear on your input and can now discuss it further. 🙂 +1 I'll come back to 'value' in the context of your introduction of 'objective v subjective' in a moment. But first your use of relativity and relationship, although these words stem from the same root, they have different meanings and usage. Even though you are not a mathematician, you should be easily able to understand the very basic concept of 'relationship' in mathematics and logic. Like so many basic concepts in so many subjects 'relationship' manifests itself as having many shades of meaning. The are many types of relationship recognised. In fact it is a broad category and we distingusih further by either introducing special new words (as in function) or additional adjectives as in equivalence relation. You might find it useful to look at your own language wiki to find out about a particularly useful one in maths called an equivalence relation. https://en.wikipedia.org/wiki/Equivalence_relation Here you can see that in this type of relation you can sometimes substitute one 'value' for another - there is no subjectivity allowed according to the rules. So subjective v objective. I hear what you say about this but Nature (Physics, maths, everything) is remarkably obstinate in resisting Man's efforts to squeeze it into his own subjective categorisations. And so it is with subjective v objective. This is not an either or (binary) choice, but rather a scale of meaning. Science, in particular, tries to remain objective by various means. We like to think that if se set up a machine to observe and/or record it is objective because it can only record what it observes. But I know that Nature can play tricks on us, from my own personal experience during my time as a surveyor. When making important verticular angular measurements it is good practice to observe from both ends of the observation line. One end will generally be below the other so looking up (+ve angle) and the other end looking down (-ve) angle. But I have seen situations where it is possible for both angles to be +ve. That is both ends of the line appear to be looking down on the other. The angular measurement instrument (theodolite) is correct (objective) and it is not operator error. So Science is able to correct its faulty theory.