studiot

Welcome. I see from this question you are studying Henderson Hasselbalch so you must already have quite a bit of simpler stuff under your belt. You also seem to indicate that you are chemistry as subsidiary to some bio or medical course. In which case this book will help greatly. The relevant section here I don't have more time this morning, but I see there are other chemists about. Also I recommend you ask a moderator to move this to chemistry since we can work through it there. Homework help is more restrictive and your question is not homework.

I think those disciplines that activeley use tensors have a head start on those which could do but usually avoid them. I wish I had had a copy of this book when I first met tensors. Geologists make extensive use of certain tensors so one can get an immediate handle on the subject via rock mechanics and, to a lesser extent, soil mechanics. The build up from one dimension is echoed in the explanation of GR given by Baez https://math.ucr.edu/home/baez/einstein/einstein.pdf and also by Koleki https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Essentially this view runs that you have a coordinate system (which is therefore geometric) and a bunch of (physical) quantities of interest at some or all points spanned by the coordinate system. This setup is organised into an array of coordinate points, an array of quantities and an array of coefficients connecting the first two arrays. The coeffeciant array being the tensor array. Its coefficients may be constants or expressions which depend upon the coordinates. When one starts some form of continuum mechanics it is usual to first work in one dimension, then two, then three and so on. This offers the opportunity to go back to a lower dimension to see what is happening, although there is always the possibility of loosing some aspect that only occurs in higher dimensions . An example of such loss is the loss of the difference between contra and co variance in one dimension. Some other books with useful visula models are from Fleisch and Bickley

I think this subject has links to your other current question about tensors so the two threads should be read together. Consequently also the references I make to in each have material relevent to the other thread. Here is a good maths text about the requirements of dimension theory, for a variety of spaces, The index of this book is particularly unusual as it contains potted definitions for many important terms. A second book I am going to place here, although it also contains good material about Hilbert Spaces it it more relevant to spaces in general than visualising tensors.

Welcome Nicely put. All good stuff. +1 Hope to see more in future.

That is not an accurate interpretation of my OP. My perception of infinite would not have an end or a final point. In what way is that not accurate ? Your very first line introuces doubts about ends of infinity. So I tried to deal with them one at a time To which I replied about the starting point Here I tried to point you at something I believe you already know. Namely that there are infinite counting numbers and you can start from any of them. Later posts deal with the end point because Infinity is so strange that you can have an infinity between a definite start point and a definite end point; or you can have a definite start point but no end point or you can have a definite end point but no start point or you can have neither end point nor start point. Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about? That is not an accurate interpretation of my OP. My perception of infinite would not have an end or a final point. In what way is that not accureate ? Your very first line introuces doubts about ends of infinity. So I tried to deal with them one at a time To which I replied about the starting point Here I tried to point you at something I believe you already know. Namely that there are infinite counting numbers and you can start from any of them. Later posts deal with the end point because Infinity is so strange that you can have an infinity between a definite start point and a definite end point; or you can have a definite start point but no end point or you can have a definite end point but no start point or you can have neither end point nor start point. Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about?

🙂 Hopefully we we wprk it out for BB in the end I admire your steadfastness so far.

The line is a space of line segments. Each of which require two pieces of information for example beginning and end points or midpoint and length. Yes it is also a space on individual points (which have zero dimension by themselves) which only require one piece of information.

How many dimensions does a (straight) line have ? Most would straightaway say 1 Yet that line is a space of elements that require two pieces of information to specify them.

Like it +1

So often in Mathematics there is more than one way of looking at things. So far in this thread we have been all been looking at what might be described as 'the assembly' method. Nothing wrong with that. But we should perhaps also consider the dissection method. I take it that you are happy that the counting process 1.2.3.4.... can be carried on indefinitely. So if you are not happy with the process of listing first, second third and placing a term in 1 - 1 correspondence with each then try this thought experiment. Take a perfect line, exactly 1 unit long Using a perfect scalpel cut off an amount exactly equal to the fraction of the line given by your first term. Call this the first slice. Then cut off a second slice exactly equal to your second term. call this your second slice. and so on and so on You know that all your slices started off adding up to exactly 1 But you will also find that you, or any being, can never reach an end to this process. Does this help? By the way, you initially complained that your series had a start and an end, against how you perceived infinity, but you never responded to me showing you that there is nothing wrong with an infinite sequence having a start and going on indefinitely.

My comment was not addressed to you and the reason why was answered in the next line. Really I am just trying to help the op get his head around the idea that assembling the infinite sequence or series is a thought experiment than cannot be completed, but nevertheless has its uses. I note that this still hasn't clicked for him despite all your proofs. I agree to no such thing. It's a philosophical question. Before the universe existed, were there numbers? Where were they? Since I have no knowledge of before the universe existed I wisely said nothing about that subject. As you say existence is a philosophical question, as is the approach I offered. Can I remind you that mathematical existence means simply that it is consistent with the adopted axioms, nothing more. Physical existance is another subject. Since you apparantly believe that numbers do not occupy zero volume (space) perhaps you can supply a formula for the volume occupied by a shadow ?

I will be interested to see the response from other member to this question but you may also find some useful source material for your lecture in this book Despite the title here is a contents list as to why it may be relevant. (Hofstadter has written some other titles which I can't comment on.)

The question is not whether the sequences 'can reach 1' but whether you can. Of course neither you nor any other being following along the sequence can ever get to the end. I hope you agree that the numbers 1, 1.5 2, 22 million along with every other conceivable number, all existed ie were available to be discovered, long before Man evolved in the universe. Luckily numbers take up zero space so there is plenty of room in our universe for all of them. So the entire sequence was there before we were and has effectively always been there.

So Wikipedia contradicts itself - so what ? you used the words dimension and space. Surely this thread is not about the definition of a manifold ? Note that Wikipedia acknowledges many kinds of manifold and dimension: affine, euclidian box-counting, Minkowski, correlation hausdorf (which I originally asked you about) amongst others. All I said was your initial specification was incomplete without further information. The fact that you are now referring to inherited characteristics demonstrates that this was reasonable. You also asked how to determine the number of dimensions of a particular 'space'. Quite a reasonable request so why are you now wasting discussion time quibbling definitions rather than discussing concrete procedures offered ?

You are correct, that was a symmetry I had overlooked. +1 Whose definition? My reference maths dictionary gives "A collection of points of a set," for a bare manifold as per Cantor's regularisation of Riemann's original useage. It goes on to add descriptive terms to narrow this down to different kinds of manifold eg for differential topology and so on. I draw your attention to the comments I already posted about set theory. In particular the 12 lines above the beginning of paragraph 96 on page 162.

I rather think we are talking at cross purposes here. My first thought in reply to the op was to note that you need more than the bare manifold to establish dimensions. In the model I originally proposed the manifold appears to correspond to the bare set of elements with one important condition. The condition that every element is of the same type. You also need at least two elements to have a metric. But this is still not enough to define or allow swansont's 'spatial dimensions'. In order to create the neighbourhoods necessary to support the maths involved you need to be able to order the elements of the manifold, ie the axiom of choice is required. Anyway I thought I was trying to discuss ways of determining the 'spatial' dimensions of the manifold when I posted my little question about non euclidian space. I wanted to get this one over quickly as it is the simplest case I could think of. So yes I am aware that there are points on the Earth, close to the poles, where one or the other journey is just not possible so to be more specific, The start point should be somwhere near the equator (anywhere will do) - I cannot see any points that would lead to the two journeys ending up in the same place. They would of course do so if the surface of the Earth was flat. This demonstrates that the surface of the Earth requires at least another dimension. Hopefully you both agree with this and we can move on to exploring what happens in more general dimensions. Much more insight is gained by studying stress paths in 1 , 2 and 3 dimensions and finding out ( as Hamilton did) that there is no 4 dimensional equivalent. I think the next dimensionality that supports this is 8, but I might be wrong about that.

I advocate the policy of going from the known to the unknown, rather than trying to guess the unknown. As such I offered a simple presentation to help you understand differential equations. Since you didn't bother to reply to my work on your behalf (your prerogative ) I feel justified in asking do you want my help or not ? I will leave you with the thought that your ratio project is generally not possible, but there are simple matters about area (and integrals) you could usefully get under your belt. It would take some drawing on my behalf.

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.