Does it make sense to take an infinitesimal volume of shape other than a cube?

4,976

Solution 1

Infinitesimal volume elements do not have to be cubes.

Some familiar examples come from typical solids of revolution problems from calculus 1/2. Typically one discusses using either the "disk/washer" or "cylindrical shells" methods for finding the volume of the solid. As you can guess, the former method uses infinitesimally thin disks/washers as volume elements, and the latter uses cylindrical shells with infinitesimal thickness.

Volumes that are finite in one or two dimension(s) and that are infinitesimal in a third dimension are still infinitesimal because an infinitesimal value multiplied by a finite value is still infinitesimal. You can also build up "non-cube" volume elements by integrating over certain variables from your "cube" volume elements. For example, you can get spherical shell volume elements by integrating over the azimuthal and polar coordinates: $$\text dV=\text dr\cdot\int_0^\pi\int_0^{2\pi} r^2\sin\phi\,\text d\phi\,\text d\theta=4\pi r^2\,\text dr$$

which as you can tell is the volume of a spherical shell of radius $r$ and thickness $\text dr$.

Solution 2

Your comments (and to a lesser extent, your Question) indicate a severe confusion about ever having an infinitesimal volume. You never construct an infinitesimal volume. Infinitesimal volumes appear at the end of a limiting process.

Where do the infinitesmial rectangular parallelepipeds you are discussing appear? They appear in the limit of an iterated triple integral. An iterated triple integral involves nested orthogonal partitions to construct Riemann sums. In the limit as the diameters of all the partitions decrease to zero, the resulting volume elements are the infinitesimal rectangular parallelepipeds you first describe.

Can there be other infinitesimal volumes? Of course; use a different coordinate system. If you have arranged your triple integral to be in spherical coordinates, then you (may, if your region of integration includes it,) have an infinitesimal sphere at the center and the rest will be volumes bounded by two radii, two longitudes (which bound a spherical wedge) and two latitudes (which bound a spherical segment). In the limit as all the partition diameters go to zero, you obtain infinitesimal versions of these volumes.

Notice that at no point during the taking of the limit do you ever have an infinitesimal volume. These infinitesimals only appear once the partition diameters finish going to zero. I'm not going to get in the philosophical difficulties of completed infinities and whether the results of infinite processes exist. The point is that we use non-infinitesimals to infer what would happen if we really could use infinitesimals.

As another example of a different infinitesimal volume, consider cylindrical coordinates. Here, we have cylinders on the longitudinal axis, and, everywhere else, volumes bounded by two (infinitely long) cylinders of constant radii, two planes of constant angle, and two planes of constant longitude. Let's give those last volumes a name: "fred"s. The cylinders and freds are not rectangular parallelepipeds. In the limit as the partition diameters go to zero, we end up with infinitesimal cylinders and infinitesimal freds.

There is a different idea -- using non-rectangular regions in the usual 1-dimensional Riemann sum. For instance, graph the function over the interval of interest, then pack the area between the curve and the $x$-axis with disks. Sum the areas of the disks. Then repeat the process in the limit as the radius of the disks goes to zero. What you find is that you do not get the same value as the usual integral. If you are careful in specifying your packing method, you will actually have a limit as the radii go to zero and the resulting total disk area will underestimate the actual integral due to the "gaps" between the disks.

In short, the method described in the first few paragraphs where we partition all of the space of integration into pieces is necessary -- don't leave gaps.

Solution 3

Different coordinate systems have different kinds of volume elements; The volume elements are a consequence of how the grid lines of the coordinate system are set. Volume element can be generated by nudging the parameters which describe points in the space by infinitesimal amounts and figuring out the volume of the region generated as a consequence. This is especially useful in multi-variable volume integrals and in the application of some vector-calculus results such as the divergence theorem.


On some more thought, I'd like to add one more point. Yes, you are correct that infinitesimals are small quantities, however you're missing a crucial point. Depending under what constraint you put while your quantity small, the actual structure of this 'small quantity' would be different. This would be understandable using the references I have given in the bottom.

As a more direct example, suppose you have a large cube and you keep scaling down the dimension till you get some sort of infinitesimal volume cube, and now for contrast, consider a large sphere and imagine scaling it down till you get a tiny infinitesimal sphere. These two things are infinitesimal volume elements but the volume each contains is different due to the actual object which you are shrinking being different.


Deriving Volume Element for spherical co-ordinates

Lecture series which shows the concept described above using 3-d animations

For Understanding the ideas of linear transformation noted in the previous lecture better


Answer V2.0 based on op's new details of the question:

  1. and 2.) You can relate the volume elements between different co-ordinate systems using the determinant of jacobian. In a way, the Jacobean is the ratio of n-volume in one system to the n-volume in another. Also do not forget that some transformations do not behave the same globally, for example it is easy to understand that the 'natural unit' vector of polar co-ordinates scales up as you move further away from the origin (*)

  2. I'm not sure on this on what exactly you mean by 'smallest'. You need an absolute measuring scale to measure the concept of smallest. If I were to guess, the smallest volume element would be a singular linear transformation which squishes space into a point and hence literally have zero volume.

  3. and 4.) Not gonna comment on hyperreal numbers as I have not done much of it and this concept was dealt with in Dave's answer in much detail already.

  1. Yes, the properties of a shape other than n-volume measures should be invariant under uniform scaling. For example, consider similar triangles.

Solution 4

An infinitesimal is by definition a length that is really, really small.

I think that your question arises due to a misunderstanding of what infinitesimals are. Infinitesimals are not easy to understand, they can be understood either as a limit as a quantity goes to zero or in terms of the hyperreal numbers. As the hyperreal concept is relatively new compared to the limit, it is not often taught, but it does have some clarity which I find helpful.

The hyperreal line is the real line augmented with infinities whose absolute values are larger than any real numbers and their reciprocals, the infinitesimals, whose absolute values are smaller than any positive real number.

The thing about the infinitesimals is that as individual numbers (not as sets) they can be manipulated with all of the same operations as reals. You can multiply an infinitesimal by a real number and get another infinitesimal. The infinitesimals can themselves be ordered, meaning that if $dx$ is an infinitesimal then $2 dx$ is larger than $dx$, but still smaller than any positive real and therefore still a perfectly valid infinitesimal.

So using “...” to denote an infinite sequence we can order the hyperreal numbers like this: $$1>\frac{1}{2}> \frac{1}{3}> ... > dx > \frac{dx}{2} > \frac{dx}{3} > ... > dx^2 > ... > 0$$ or more colloquially we can consider $\epsilon =0.000...1$ to be a sort of unit infinitesimal which can still be divided by 2 to make something even smaller and so on. There is no absolute smallest infinitesimal number. As an exercise, consider $dx$ and $\epsilon$. Which is smaller$^*$? Is $dx<\epsilon$ or $\epsilon < dx$?

This is important because infinitesimals can preserve their relationships to each other. They are all smaller than any positive real, even if some infinitesimals are larger than other infinitesimals. So $dx \ dy \ dz$ is half the volume of $dx \ dy \ (2 dz)$, but they are both infinitesimal.

In fact even if $x$ and $y$ are finite real numbers $ x \ y \ dz$ can be an infinitesimal volume. An infinitesimal volume merely needs to be smaller than any positive real volume, not smaller than other infinitesimal volumes. For that a single infinitesimal in the product suffices. A spherical shell from radius $r$ to $r+dr$ is a completely legitimate and valid infinitesimal volume $4 \pi r^2 dr$ even though its surface area is finite $8 \pi r^2$. This all follows from the properties of hyperreal numbers.

Infinitesimals can be formed into a hyperreal plane and into vectors, and those vectors can have norms and dot products, so you can have arbitrary infinitesimal shapes. You can have right angles, but you can also have arbitrary other angles. There is nothing magical about right angles that allows them and forbids other angles. You can have straight lines, but you can also have arbitrary curved lines. There is no restriction to right angles and straight lines.

Since you realize that infinitesimals can be orthogonal to each other, it should not be surprising that there is no limitation to other angles and thence to arbitrary shapes. The same rules that allow you to construct orthogonal infinitesimals allow you to construct other shapes. Again, all of this follows from the hyperreals.

Is not the infinitesimal cube the absolute smallest infinitesimal volume?

Responding to this most recent aspect of the question. There is no absolute smallest infinitesimal volume. You can always make a volume smaller.

For instance if $dx \ dy \ dz$ is an infinitesimal cube then we can define $dx = 2 dX$ and then $dX \ dy \ dz$ is a smaller volume and is not a cube. Similarly, we can define $dx = 2 dr$ and then $4\pi/3 \ dr^3$ is an infinitesimal sphere which is smaller than the cube. And simply by using a bigger number than 2 we could make volumes smaller than those. There is no absolute smallest infinitesimal volume.


Since many people are not familiar with hyperreals, here are some introductory sites (by no means complete or optimal):

https://www.youtube.com/watch?v=FTXRnEKEn4k

https://en.wikipedia.org/wiki/Hyperreal_number

https://www.youtube.com/watch?v=ArAjEq8uFvA

http://mathforum.org/dr.math/faq/analysis_hyperreals.html

http://homepage.divms.uiowa.edu/~stroyan/InfsmlCalculus/FoundInfsmlCalc.pdf


$^*$ In this case $\epsilon < dx$. Notice that $dx$ is defined by: $$ 1 > \frac{1}{2} > \frac{1}{3} > ... > dx$$ and $\epsilon$ is implicitly defined by: $$ 1 > \frac{1}{10} > \frac{1}{100} > ... > \epsilon$$ Since every term in the second sequence is smaller than the corresponding term in the first sequence $\epsilon < dx$

Solution 5

It’s not so much a question of what is theoretically correct, more a question of which shape of region allows us most easily to pass to the limit and derive a differential equation or an integral (which is usually the goal of this step).

The choice of region often depends on the symmetry of the problem. In problems with cylindrical symmetry it is common to use a cylindrical shell. In problems with spherical symmetry a spherical shell is often used.

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Sidarth
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Sidarth

Updated on August 01, 2022

Comments

  • Sidarth
    Sidarth over 1 year

    The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume?

    (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything other than the Cartesian volume element?" : I know the answer to this is of course yes and I know it's usefulness. But please note that the question title has not been changed at all! It stands.)

    After the many discussions, now the questions stand at comparing infinitesimal volumes.

    A holistic answer that addresses this will be appreciated. This involves a phrasing of what infinitesimals are, how an infinitesimal volume arises, and what happens when such volumes are compared from two different coordinate systems. Is it OK to address the infinitesimal volumes as smaller versions of finite shapes? If it is fine, what is wrong in this Gedanken?:

    • An infinitesimal is by definition a length that is really, really small. If then I multiply this length with the same but in two perpendicular directions, I get a cube. This is the infinitesimal volume in Cartesian coordinates.I.e., an infinitesimal volume should have all it's edges as infinitesimal lengths, right? Is any other infinitesimal volume theoretically correct? (I have trouble accepting cuboid-shaped "infinitesimals" as well.)

    I would highly appreciate people from a physics background to answer this question in an intuitive, "Feynman lectures" way, for lack of better words. Everyone's time is appreciated!

    My argument for the comparison of volume elements across different coordinates systems:

    In any coordinate system, I can define an interval whose unit length I can define, right eg. $|ds|=\sqrt{dx^2+dy^2+dz^2} $. So, the infinitesimal volumes from any coordinate system can be compared. Given this and that infinitesimal volumes occur, as a result, I would start off with 3 infinitesimal lengths with no possibility of a curved surface. I should end up with a cube only.

    • Qmechanic
      Qmechanic about 3 years
    • Buraian
      Buraian about 3 years
      I think your question would be much easier to answer if you could boil it down to a few points which you didn't understand. As of right now, it is a large mess after the amount of edits you have made
    • tpg2114
      tpg2114 about 3 years
      Comments are not for extended discussion; this conversation has been moved to chat.
    • BioPhysicist
      BioPhysicist about 3 years
      Please make your question one cohesive post instead of tacking on edits at the end. An edit history is available for those who are interested.
    • Buraian
      Buraian about 3 years
      Much better but if you could summarize each passage into a bullet point list, it'd be much easier to understand. It takes a lot of mental effort to sit down and work through each paragraph when the discussion is not about something 'concrete'.
    • Buraian
      Buraian about 3 years
      However I'll try to edit and add to my answer, check it out in a while
    • Fattie
      Fattie about 3 years
      "An infinitesimal is by definition a length that is really, really small." You can't really start off epistemological discussions with assertions like that. On top of that, it is a vague/incohesive assertion.
    • chepner
      chepner about 3 years
      Is there a physical definition of "infinitesimal" that I'm not aware of? Who said it has to be a length, rather than a pure, dimensionless, numerical value?
    • Sidarth
      Sidarth about 3 years
      @Fattie This question was aimed at understanding something that I did not understand. I can only word the question with things I know of. For me, infinitesimal is the smallest length I can think of. (Just as how infinity is something that is boundlessly large)
    • Sidarth
      Sidarth about 3 years
      @chepner we need dimensions of length for the triple integral to be a volume
    • Dale
      Dale about 3 years
      So did you just let your bounty go to waste? That is kind of bad form here
    • Sidarth
      Sidarth about 3 years
      @Dale I missed the deadline by an hour! Sorry about that.
  • Sidarth
    Sidarth about 3 years
    Non-euclidean will be tough for me. I am trying to visualize it Todd. Qmechanic pointed me to a source that talked about this with mathematical sophistication but I got lost. I said cuboidal. Not paralleopiped. Your explanation about the smallest volume is very useful for my question - the arbitrariness - but it sounds more to do with numerical methods and accuracy rather than analysis - which there is some absoluteness. Infinitesimal means smallest, right?
  • Todd Wilcox
    Todd Wilcox about 3 years
    @Sidarth Infinitesimal is a special term of art used in calculus to mean a mathematical object that has certain properties. In this context, it is not helpful, in my humble opinion, to reduce the definition of the word infinitesimal to "smallest". I know you said cuboidal. I was giving you an example of a non-cuboidal infinitesimal (a parallelepiped) that is used in a very famous book. In my answer, the word "analysis" means, essentially, "calculus". Calculus is basically numerical methods (the mathematical topic) with limits (the mathematical concept) applied.
  • Sidarth
    Sidarth about 3 years
    So, calculus finally comes down to getting a numerical answer so the arbitrariness of the smallness of the elemental volume depends on the problem, is that what you are saying?
  • Todd Wilcox
    Todd Wilcox about 3 years
    @Sidarth I think the "shape" of the infinitesimal depends on the problem, and the "shape" is created by the linear transformation that defines the infinitesimal, which also depends on the problem. How small (or large) an infinitesimal can be made to be does not depend on its shape. We can make all kinds of shapes arbitrarily small. In some cases, it's harder to use a cube/cuboid than it is to use a different shape, which is why we don't always use cubes/cuboids.
  • Sidarth
    Sidarth about 3 years
    I had to read your answer multiple times to understand! It might be the way that I have been schooled that I have never thought of infinitesimal volume elements as the "end result" of anything. So, I take a coordinate system first, assign a shape in the central region (shape is obvious?) and then any volume is like a "skin" above it. An infinitesimal volume occurs when the grid lines cut into this thick skin, then make the grid lines take the limiting value. Is this correct?
  • Eric Towers
    Eric Towers about 3 years
    @Sidarth : Yes -- you don't have infinitesimals until you reach the limit of infinitesimally separated grid lines.
  • Ruslan
    Ruslan about 3 years
    "iterated triple integral" — so iterated or triple? These are distinct objects.
  • Eric Towers
    Eric Towers about 3 years
    @Ruslan : "Iterated" does not by itself indicate the depth of the nesting. In the OP"s case the depth is always $3$ as indicated by "triple". When Fubini applies (which will be the case for anything OP is attacking), the non-different result of the non-distinct notation makes the distinction you are drawing a distinction without a difference.
  • Ruslan
    Ruslan about 3 years
    When Fubini theorem applies, "iterated triple" is simply redundant.
  • Eric Towers
    Eric Towers about 3 years
    @Ruslan : Repeating what I said doesn't change anything.
  • Ruslan
    Ruslan about 3 years
    My previous comment was supposed to mean that "triple integral" is the non-redundant term that does indicate depth.
  • Eric Towers
    Eric Towers about 3 years
    @Ruslan : Iterated integrals have the nested partitioning described in the Answer. Did you not follow that?
  • Sidarth
    Sidarth about 3 years
    "...An infinitesimal volume merely needs to be smaller than any positive real volume, not smaller than other infinitesimal volumes...." I did not know this! Very non -intuitive since I was initially thinking that the infinitesimal volume has to be vanishingly small, meaning there is only one possible infinitesimal volume, one made of infinitesimal lengths.
  • Dale
    Dale about 3 years
    @Sidarth I am glad it helped. I have only recently learned of hyperreal numbers and now wish that calculus were taught using them
  • Sidarth
    Sidarth about 3 years
    Yes. Definitely it has helped. But I also feel that an extra structure has been imposed in order to make something that was initially non-intuitive seem more plausible. I feel that it did not address the problem straightforwardly but rather justified somethings by using extra definitions. (I am not a mathematician. So this approach is a little different to me!)
  • Dale
    Dale about 3 years
    @Sidarth yes, I agree. But structures that make initially non-intuitive things seem plausible are my kind of structures! Particularly if the new structures are consistent with the established but non-intuitive approach, and are easier to use. The extra definitions are then worth learning. Of course, that is pure personal opinion and you are under no obligation to agree!
  • Sidarth
    Sidarth about 3 years
    Right. I think it's safe to say that I am completely in rigorous mathematics now?
  • Buraian
    Buraian about 3 years
    Could you give some further readings/ citations on this answer? I found it quite interesting
  • Dale
    Dale about 3 years
    @Buraian excellent suggestion! I have added several links to the answer. By no means is that a complete list, but it is roughly organized in order of increasing rigor and decreasing accessibility.
  • Buraian
    Buraian about 3 years
    If you could write meaning of these mathematical terms you use as a footnote or smthn this would be an easier read
  • Buraian
    Buraian about 3 years
    If you define measure and measurable sets, this would be easier to understand for a non-math person and also what a Jordan domain is. If not, add some references to what you're saying.
  • Buraian
    Buraian about 3 years
    What do you mean when you say no local structure? I think you'd help op if you got into geometry of volume
  • Buraian
    Buraian about 3 years
    I wish this answer got more attention, it displays a nice idea!
  • Buraian
    Buraian about 3 years
    I don't understand point-2
  • Sandejo
    Sandejo about 3 years
    @Buraian I added some links and explained some terms.
  • user52817
    user52817 about 3 years
    @Buraian: A good example of local structure in Riemannian geometry is curvature, which is a local invariant. Perhaps a familiar example of a geometry that has no local structure is symplectic geometry. This follows from Darboux's theorem. Moser's 1965 paper gives an excellent proof. Suppose one person has conviction that infinitesimal cubes are "the volume element" while another person has conviction that another geometry is, e.g., the spherical volume element. Both need to accept that there exists a local volume preserving diffeomorphism converting one viewpoint to the other.
  • Ryan Thorngren
    Ryan Thorngren about 3 years
    I'm not sure this qualifies. The basic volume element is still a prism of dimensions $r d\phi$, $r \sin \phi d\theta$, and $dr$.
  • Kevin Arlin
    Kevin Arlin about 3 years
    Re 1: Sure, why not? In 2D polar coordinates a volume (area in 2D) element is a sector of an annulus, and its area at $(r,\theta)$ is $rdrd\theta$. If $(r,\theta)$ has the rectangular coordinates $(x,y)$, then the area of the corresponding volume element is $dxdy$. Which is bigger? Well, it depends on what $r,dr,d\theta,dx,$ and $dy$ are! If you choose $dr,d\theta,dx,dy$ to all equal your favorite infinitesimal, then the polar area element is larger than the rectangular when $r>1$ and smaller when $r<1$.
  • Kevin Arlin
    Kevin Arlin about 3 years
    It's basically unavoidable, by the way, that an infinitesimal not be "the smallest possible length", if you want to preserve any connection with differentiation. Here we say things like $(x+dx)^2\approx x^2+2xdx$" to calculate the derivative of $x^2$, but to do even this we must be able to multiply an infinitesimal by $2$! Thus we must have another infinitesimal which is twice as big...And what if we divide it by 2? The hyperreal perspective has the benefit that the hyperreals have all the same basic properties as the reals, just plus some infinitesimals. So your comment after (5) is right.
  • Sidarth
    Sidarth about 3 years
    @BioPhysicist I do know that cubes are not the only infinitesimal volumes. "... because an infinitesimal value multiplied by a finite value is still infinitesimal..." then the discussion moves to comparing infinitesimal volumes.
  • Sidarth
    Sidarth about 3 years
    1) Could you explain how this is going to affect the infinitesimal volume:"...Also do not forget that some transformations do not behave the same globally..." 2) In any coordinate system, I can define an interval whose unit length I can define, right eg. |ds|=\sqrt{dx^2+dy^2+dz^2} . So, the infinitesimal volumes from any coordinate system can be compared. 4) Give this and that infinitesimal volumes occur as a result, I would start of with 3 infinitesimal lengths with no possibility of curved surface. I should end up with a cube only.
  • Sidarth
    Sidarth about 3 years
    The first definition is very useful and is what I picture of as well. The second part of the answer is correct of its own accord. But it does not help me. Sure, I have defined an infinitesimal volume but I have lost the ability to compare it with the cube.(right?) The highly voted answer does not really completely answer my question that is why I have not ticked it. The question is still there. Please contribute in your own different way. I would love if someone could explain to me without things like for e.g. the mathematical "tricks" eg, with new definitions to justify what is being done.
  • Sidarth
    Sidarth about 3 years
    1st comment) @KevinArlin This is the first time I'm pointed to the size of the infinitesimal volume itself changing as the place I take it changes! i.e., the r coordinate. '...favourite infinitesimal..." Do you mean assign a numerical value irrespective of the fact that angles are dimensionless? So the polar case is actually smaller than the Cartesian one in r<1 case. It might become still smaller in the 3d case because of another r dependence.
  • Sidarth
    Sidarth about 3 years
    w.r.t 2nd comment) Then, I want to compare the hyper real infinitesimals.Not infinitesimals with finite things. I don't have a problem with an infinitesimal being multiplied by 2 and being called an infinitesimal. That just refers to the fact that we are dealing with small things compared with our integration limits(?) .In the hyper real infinitesimals also, I can have the smallest value - the "infinitesimal even in the hyper real numbers". Then I can use this. (I hope no one says there is something now called hyperhyper real numbers!)
  • BioPhysicist
    BioPhysicist about 3 years
    @Sidarth Then ask a new question. Just make sure to be specific about the comparison you want.
  • Sidarth
    Sidarth about 3 years
    I have added the final edit to the question which points this out. Should I ask a new question? What about the activity on this one? Frankly, comparing infinitesimal volumes has been the question all along.
  • BioPhysicist
    BioPhysicist about 3 years
    @Sidarth If your own answer is what you are looking for then you can accept it. I feel like the goal posts have been changing for this question for a while. Technically if you realized you were wanting to ask a different thing you should have asked a new question from the beginning, not update your question so that current answers became somewhat invalid.
  • Sidarth
    Sidarth about 3 years
    @BioPhysicist I've re-read the question many times. The question remains. I still don't have a satisfactory answer. Now I am in the process of comparing infinitesimal volumes. (which actually is the original question), now, its only clearer after filling some gaps about how exactly an infinitesimal volume is formed, the concept of hyper real numbers, some knowledge about infinitesimals itself. It seems I can get a yes or no answer right? Please see my unaddressed comment to Buraian's answer as well
  • BioPhysicist
    BioPhysicist about 3 years
    @Sidarth I answered your question a while ago before you drastically changed the question. This is poor practice because you are invalidating answers that have already been made. This site isn't a "back and forth" forum where you keep updating and changing goal posts and then expect answerers to do the same. I honestly can't even tell what you are asking anymore. Your question has become very unfocused, and I don't want to dig through comments to try to figure out what you're really after here.
  • Neil_UK
    Neil_UK about 3 years
    @Buraian so if it mattered whether you took the lefthand side or the righthand side of the infinitessimal volume, then how would you get to the integral by taking the summation to a limit?
  • Sidarth
    Sidarth about 3 years
    I don't take your time and efforts for granted. Thank you.
  • Bruce Lee
    Bruce Lee about 3 years
    +1, correct answer and addresses the question completely IMO.
  • Dale
    Dale about 3 years
    @Sidarth I added a section addressing your current question.
  • Deschele Schilder
    Deschele Schilder about 3 years
    This is not really explained in a "Feynman lecture" way. -1
  • Sandejo
    Sandejo about 3 years
    @DescheleSchilder Does it need to be? That part of the question was added after I wrote this answer.
  • Sidarth
    Sidarth about 3 years
    @Dale Hey Dale. Your recent addendum seemed spot on then I recollected my own doubts (see OP : .."I have trouble accepting ...") but I am not denying this. It is perfectly acceptable that I can go to smaller volumes given that the infinitesimals are "ordered". This seems to be like the playing around the concept of "boundlessly small" - no matter where I am, I can go smaller. One thing I have trouble :in dXdydz, dX is smaller than the other two "infinitesimals". Now, "infinitesimals" is just a definition and lost it's meaning.Any comments?
  • Dale
    Dale about 3 years
    @Sidarth “Just a definition and lost it’s meaning”. The definition of a word IS the meaning of that word. It didn’t lose any meaning, it just never meant “absolute smallest” to begin with. I added a paragraph above about the ordering of the hyperreals that should help see why there is no absolute smallest
  • Sidarth
    Sidarth about 3 years
    I understand that there might be no "absolute smallest" @Dale, but in a given problem, when I am solving, in a given line, when I write a product of infinitesimals, keeping in touch with the word infinitesimal as boundlessly small, I would not want to take one term (which is boundlessly small) in the product as "lesser" than another boundlessly small quantity , right? (Hope this is not irking you!). Also, its very acceptable that there is no one absolute smallest thing.
  • Sidarth
    Sidarth about 3 years
    I am reading this answer.
  • Dale
    Dale about 3 years
    @Sidarth "I would not want to take one term (which is boundlessly small) in the product as "lesser" than another boundlessly small quantity , right?" I don't know how many ways I can devise to tell you that yes you can and often do take one boundlessly small quantity as smaller or larger than another boundlessly small quantity. That is exactly what it means for the infinitesimals to be ordered. Here is a question for you: Given the definitions of hyperreals, in the paragraph I added, which infinitesimal is larger $dx$ or $\epsilon$? They are not equal, one is definitely smaller than the other
  • Sidarth
    Sidarth about 3 years
    "That is exactly....to be ordered." - I agree. I think what I am doing is questioning whether then such a definition is "correct" (physicist point of view) but then again, mathematical definitions don't care about physical reality or daily parlance ( since this argument on the other end of the number line would be like comparing different infinities and saying one infinity is greater than the other, even though infinity means something that's bigger than the biggest number I can think of)." I don't know how.... ordered." - you sound like an exasperated professor. Love it :) Thank you ,sir!
  • Dale
    Dale about 3 years
    @Sidarth you said "I think what I am doing is questioning whether then such a definition is "correct" (physicist point of view)". You already know the answer to that. In your own words: "I know the answer to this is of course yes and I know it's usefulness." From the physicist point of view it works, as you already know. There is nothing more to it from the physics point of view. FYI, the hyperreal infinities are also ordered, just like the infinitesimals.
  • Sidarth
    Sidarth about 3 years
    Gyro, the problem is not "what is an infinitesimal" but rather "can I talk about the product of infinitesimals that are unequal". I take infinitesimal to be a boundlessly small length - it has no fixed value and is undefined. We denote that as dx (say). Nothing can be smaller than this. Now, how can I then, in the same problem define d$\alpha$ , which is smaller and one more and multiply these and call it an infinitesimal volume.That's the point. But @Dale has given some explanations w.r.t this,viz the ordering in infinitesimals and how "infinitesimal" does not really mean boundlessly small.
  • Sidarth
    Sidarth about 3 years
    I agree that at first I did not consider that the volume element changes with where it is taken in other coordinate systems. But since I am considering the smallest volume element in that system as well, I would ensure I get the smallest thing possible (algebraically, not quantitatively) in that system then compare with the infinitesimal cube. w.r.t your 3rd paragraph: the point is not the "quantitative" infinitesimal volume. Once a "boundlessly small" a.k.a infinitesimal exists, you can't give it a number. Giving it a number is suited for numerical analysis but my question is not that.
  • Sidarth
    Sidarth about 3 years
    Great line about the Planck length. Brings some physics in. "...Don't you think a tetrahedron..."- I just checked out the equilateral square pyramid: It satisfies my condition for using only one length - the absolute infinitesimal and it's volume is lesser than a cube! Yes! I can define a grid system whose axes are along this pyramid. I don't think there is a problem and I guess the infinitesimals will be equilateral square pyramids.
  • Sidarth
    Sidarth about 3 years
    Suddenly, I am under the impression that any infinitesimal volume element other than a sphere is wrong. It satisfies the condition that I use only one length. The square pyramid also satisfies the one length criterion and has much smaller volume.Still.