joigus

Good insights, guys. (+1 to both) For @grayson,

The best approach to a tensor, I think, should be dealing first with Euclidean tensors. Ie, fixed entities that represent multilinear mappings acting on vectors on a flat space. On a flat space we can identify points with vectors quite directly by means of an affine structure. On a manifold OTOH, points are not vectors, nor can they be identified with such. We must introduce vector structures point to point (the so-called tangent space at point x TM(x) M standing for "manifold"). So I would introduce tensors in two steps. First: What they do at a point; then considering how what they do changes from point to point. That naturally leads to a parallel transport as a rule to take vectors at one point to vectors at a different point. Flat spaces have no connection (or parallel transport), even though they have tensors. Another thing that confuses people is the difference co-variant / contra-variant, which occurs long after one finds tensors and has to do with vectors "naturally" having two alternate bases when we are not in an orthonormal frame. Trying to bring all these aspects together into one pictorial explanation is, perhaps, misguided. I don't know. The root "tens-" in tensor doesn't help either. It seems to suggest something directly physically interpretable having to do with tension. They are multilinear operators; that's what they are. Angular velocity is a 2-tensor on a flat space, but in disguise. x-posted with @studiot

This is actually a question that's very close to my heart, so I'll be looking forward to derivations into both pure mathematics as well as physics, as Swanson and Hanke have suggested.

I, for one, don't feel any need to visualize a tensor. Never have, I must say. To me, it's an algebraically motivated concept. In some cases it might be useful to picture something geometric going on (example: the energy momentum tensor in GR). As an example of the converse, the Einstein tensor is very notorious for being essentially the only second-order tensor you can form that's covariantly constant. I'm sure pure mathematicians will tell you that there is no known role that this tensor plays in pure geometry --unless you invoke GR for some reason.

Oh, OK. I was confused by your words "you are given a space of unknown geometry". You must be given something. You must depart from some assumption. I think that may be at the root of why some of us have thought you were talking about physics. I've been known to do that...

Can you actually provide the example? Otherwise it's like... "I'm thinking of a space... you know. It's fractal here and non-fractal there, non orientable...", and so on. You see what I mean? Which one is it? Meanwhile, in coordinate land, say... f1(x,y,z)= x+y2-1=0 f2(x,y,z)=2x+y-cos(xyz)=0 The implicit function theorem guarantees there are ranges of x, y and z that make this a 2-dim analytic manifold, at least for certain values of x, y, z. Then you can worry about metric, angles, and parallel transport. Thus we would have a metric space with parallel transport, so we can talk about infinitesimal distance, angles, and distance along a path. But dimension is there at the very beginning.

Skimming through this very interesting topic, with special emphasis on Genady's quote from MTW. The first thing that came to my mind is that @Genady's question is, I think, equivalent to, Is there any way to define dimension --in pure mathematics-- that can be considered more primitive* than counting coordinates? A connection requires a differentiable structure. For that you have to have to be given your space in terms of equations, whether implicit or explicit, parametric, etc. From there, derivatives allow you to give a sense to the concept of "moving" (vectors). The so-called tangent space. Metric and parallel transport (connection) can be introduced independently. In spaces given in this way, you can start talking about dimension long before you have a metric or parallel transport. Eg, in thermo you have systems defined by eq. of state like f(p,V,T)=0. Even though you don't have any meaningful metric, or parallel transport (although you could talk if you want about a tangent space consisting in the different thermodynamic coefficients); you do have a dimension, which in the example is = 2. MTW's criterion is, I think, based on the topological notion of space. A topological space is basically a set with an inclusion operator on which you can define interior, exterior, and boundary. This I have found in https://u.math.biu.ac.il/~megereli/final_topology.pdf, which in turn I've found by googling for "dimension of a topological space". It seems that this criterion is somewhat different from MTW's, but they agree in that they're both topological. I must confess I'm a tad out of my depth with these "coverings" and "refinements of a covering" in topology. I must also say I'm always baffled by these questions when no analitic example (ie, using coordinates) is allowed. How are you even given a space when no coordinates are allowed? * Relying on fewer assumptions, that is.

You don't make sense. More examples: (My emphasis.) This is like saying that Einstein, rather than being a physicist, was German. Thus, whether something is a field, or a high-dimensional state --of what, BTW?-- belong in different categories. And more: (Again, my emphasis.) Blend QFT with ST?! You seem to forget that when people say "string theory" that's just short for "supersymmetric quantum field theory of strings". So string theory is but one kind of quantum field theory. Again using analogy, what you're saying here is very much like saying "we should blend calculus and mathematics". It's obvious to most everybody here that you're not making any sense. You've found a narrative that pleases you in terms of these characters "entropy", "mass", and so on. That's not science.

Can you "bunk" it first? It hasn't escaped my attention that you've used the word 'therefore' incorrectly three times since you've been here.

Sounds like an appealing approach to physicists and engineers. Thank you, Studiot. Although I've been much less active lately, I was also wondering about Markus. I found I'd missed a brief message announcing he was to be away for a while. I hope he's OK too.

There's a reason why science fiction is called fiction. The property you want to circumvent is called "confinement" of the strong nuclear force. It would be interesting to try and see if people have thought about this. I bet it's impossible, but that's never stopped people from trying.

You have to learn to distinguish everyday language from technical language. As @CharonY said, "evolution" means something different from the use in sentence "slang evolves over time". Slang does not evolve over time; it's just replaced by a different slang, for reasons well explained in quote by @Genady. Slang doesn't evolve over time in the sense that species evolve, or memes evolve (in the sense proposed by Dawkins). And that's just swell.

It is possible that you mean the differential of a function. As @exchemist said, taking the derivative and differentiating is used synonymously. But sometimes people talk about the differential, or to differentiate the variable / function, in the sense of taking small increments, and then using the derivative. \[\Delta y = \frac{dy}{dx} \Delta x \] https://en.wikipedia.org/wiki/Differential_(mathematics)#Introduction Mathematicians are more rigourous, and would say that values of a function close to a given point can be expressed as a linear function of the local values of the independent variable plus a small increment: https://en.wikipedia.org/wiki/Differential_(mathematics)#Differentials_as_linear_maps I hope that was what you meant.

No. It means f depends on x. So you might have f(x) = 2*x or f(x) = x2 etc.

Sorry, I meant "the derivative of x4 at x=5 is indeed 4*53" I hope that was clear... If it wasn't, please tell me. I'm sure you can.

No. It means you can't take x to be 5, or any other particular value. It must be a variable (varying, non-fixed) quantity. So the derivative of x5 is indeed 5x4, while 4*54 'is nothing of' 55. And the derivative of x5 at x=5 is indeed 4*54 But that is not what you said... It seems as if you're getting ahead of yourself. Maybe you need a good calculus book --like Spivak--, instead of calculus for dummies.

That must be it.

There is no evolutionary point in any change. Evolution comes into play when changes become stable through time. Today's slang is not the same as 18th-century slang.

Doubtful. That is plain wrong.

Looks like you're having trouble with the properties of powers, not with antiderivatives. Are you familiar with xn+m=xnxm? x-posted with @Genady

This is pseudoscience.

Very good questions indeed. The only proof of a scientific theory is experiment. Theory by itself doesn't allow us to prove a theory right, but it does allow us to prove it wrong. My advice would be to try to master trigonometry and calculus first. Also physics and chemistry, of course. Then algebra, geometry, topology... the works. Quantum mechanics, relativity --both special and general--, quantum field theory. Once you understand general relativity and quantum field theory, it's possible to understand why superstrings are perhaps worth considering. That's the pathway in a nutshell. You're allowed to enter a 'room' before you've completely understood the contents of the previous one. Otherwise it would take several lifetimes.

Sure it is. In fact, given enough context, a complete reference to the terms could become unnecessary, and saying 'I see' or 'I don't think so' could be enough to make clear what one means. If you think about it, we use elipsis most of the time when we are with family or close friends. They know what we mean. When I said syntax must have been present very early on, I meant that even in the first stages of development of language there must have been a very simple set of rules (subject)+(action)+(object) --or inverse order--, (subject)+(be)+(attribute). I don't think that crude pointing at things and naming could have been going on for much long. Linguists call this proto-syntax. It is, no doubt, speculation on the part of linguists --as language leaves no fossils--, but a very reasonable one. So obviously all words are more words couldn't be farther off the mark.

Or said index finger, but only when it's pointing to a rock. I think syntax is necessary very early on.