Related: How to treat differentials and infinitesimals? , Rigorous underpinnings of infinitesimals in physics and links therein.

You might work in polar coordinates. Try to calculate the volume of a sphere by integration.

@Qmechanic So, I take it that the changes in the infinitesimal volumes when taking infinitesimals of different coordinate systems, it is a higher order correction and can be safely discarded? badjohn, I know that but I think it's more to do with is it actually right to say that more than one infinitesimal volumes can exist .

In general relativity one sometimes encounters so-called “small-sphere limit” for various quantities (like energy) at a point. See e.g. gr-qc/9810003.

What do you mean by higher corrections?

@Buraian I read Qmechanic's links the best I could. What I could infer was some neglecting of some higher order terms. I accept however that even I am unclear as to the exact place where we get the higher order term and can neglect it. Perhaps you can take a look as well and explain. Would help!

Here goes: Cube has three lengths, all equal (since nothing can be smaller than an infinitesimal length). Another shape is the elemental volume of a SP coordinate system. Is the volume given by this object the same as that of the elemental cube? If not , is the error low enough so that it can be taken as an approximately infinitesimal volume element? (perhaps the volume correction involves a product of 4 or more of the infinitesimal lengths) . (Sorry I am not pointing to the exact place from Qmechanic's link)

Something I don't quite understand from your question: What do you mean by "the smallest possible volume"? Isn't the smallest possible volume 0? It seems like there's a reason why you think that the different edges of a "shape" used as an infinitesimal should be "the same size", but it's not clear what that reason is. All that is needed is that all of the edges can be scaled by a known linear transformation so that the size of all of them is arbitrary and fixed. If you had a prism with sides $t$, $t$, and $3t$, you could use that, since you can scale by $t$.

What do you get if you make a sphere with infinitesimal radius?

"since nothing can be smaller than an infinitesimal length" - I'm not sure what you're talking about here. There is no smallest non-zero length. If you give me any non-zero length, I can give you a smaller one. There is no abstract size or length that is "an infinitesimal size/length". The volume of an infinitesimal need not be "the same as that of the elemental cube". We do not require that to be true to do calculus.

@AsteroidsWithWings I cannot use that as an elemental volume, right? How is this helpful?

@Sidarth Okay sorry

You absolutely can use a sphere as an elemental volume. Why do you think you cannot? I mean, it doesn't exactly make the math easy, but it doesn't make it impossible. Ok the more I think about it the more I think it makes the math not just difficult, but intractable. But the concept is not ridiculous. The problem is that it's very hard to count the number of spheres that fit inside an arbitrary volume. But as an infinitesimal, it's has two salient qualities: Scalable and of known, fixed volume.

@ToddWilcox "... I can give you a smaller one..." : Given a problem, I fix what is an infinitesimal length - a numerical value ,even. Having done that, I am only allowed to use the magnitude of this infinitesimal length in the other two dimensions . If I go smaller, that breaks my definition of infinitesimal length , right? (Note that fixing what an infinitesimal length is for a particular problem is something you have rubbed off on me! I did not have that idea in my mind initially. For me, infinitesimal means boundlessly small - which applies to all problems and is not a number)

"I fix what is an infinitesimal length. Having done that, I am only allowed to use the magnitude of this infinitesimal length in the other two dimensions" - I'm not sure where you're getting that from but I think you're on the wrong track. No calculus book I've ever studied has anything like that in it. Note that, in math at least, "boundlessly small" means there is no smallest member.

I did not get it from anywhere. It seems intuitive, right? THe very next sentence justifies it. What is wrong in that?

If you came up with that yourself, then I suggest you might be doing calculus differently from how everyone else has been taught to do it. So you might not get helpful answers from others about a calculus that you have invented for yourself. Regarding the next sentence: "If I go smaller, that breaks my definition of infinitesimal length, right?" I suggest you use a different definition of infinitesimal length. I notice you called it "my definition". I suggest you find a definition formulated by someone else and work to understand and use that definition.

Please don't take it as my definition, I did not enforce the myness - initially, I took "infinitesimal" to be boundlessly small, then after discussing with you, I thought about it in a more numerical method perspective and fixed an actual value to it. So, finally to define what an infinitesimal even is: "infinitesimal" is problem dependant and is not absolute?

I suggest that you reread the answer and see that 3-d link

When you impose a co-ordinate system onto space eg: paralelly space grids lines for cartesian and concentric circles + lines of constant theta in polar, you sort of are choosing how you the area looks like in that space

However I think your question is more about a co-ordinate independent concept. For that I say that like it's just like shrinking down something... suppose you took a pencil and shrank it down smaller and smaller, you could possibly get an infinitesimal pencil. As you scale an object down, the shape of the smallest object resulting from the scale down operation is dependent on what object you started scaling down by

@Buraian "...you sort of are choosing how you the area looks like in that space.." I guess this is where I have to abandon a physical volume and go into mathematics and the definitions of what a volume or area is , given a coordinate system - where we fix something called an infinitesimal length, then apply transformations to go to a different coordinates, due to which we have to transform the volume/area elements correspondingly as well, which may turn out not to have the same magnitude as the volume before (if I plug in numbers)

Gyro's answer and your links have been a great help and I have seen the links.

well no, there is no real preference to any particular co-ordinate system where we say the infinitesimal length in one co-ordinate system is the best. Tensors deals with this problem very well, I suggest that you search up the tensor calculus playlist on youtube by "maththebeautiful"

The area element is a property of gridlines that you instill on your space, and, the area of one co-ordinate system has 0 bearing on how the area is another. However, there are conversion formulas for the area in one co-ordinate system to area in another. Helps in integrals.

I think the easiest physical example way to think of it using window wipers. If we have a straight window wiper free to rotate about a point, then we can only wipe like circular areas, but suppose we could slide the wiper across the window, then we could sweep a rectangular area

Is there THE infinitesimal volume element? No. It is a dynamic approximation of something getting arbitrarily small. $\epsilon >0$.

I wouldn’t say “fix an infinitesimal length”. The shape of the integral volume is fixed, but the size of it is not. It’s arbitrary. The core process of calculus is varying the size of the infinitesimal, taking the limit as the size goes to zero. That’s not fixed.

@GyroGearloose What is $\epsilon$ that you are referring to?

@ToddWilcox by "integral volume" , do you mean the elemental volume? If so,do you mean it is scalable, as per our wish? and what do you mean about the shape being the same? The Cart.coor elemental volume is visually different from that of the SP coor elemental volume.

I think he is using the mathematician definition. In mathematics we refer to the generalized measure as n-volume.. for example: length = 1- volume, area = 2-volume, volume=3- volume etc.

@Sidarth is the famous $\epsilon>0$, used traditionally in definitions of limits, especially in the definition of derivatives. It is a positive variable that is especially interesting if it gets smaller and smaller; infinitely small, but does not have any fixed value. Question: what is the smallest positive real number you know? Yes, there is none, because you always can take half of it.

@Buraian "...I think he is using the ..." Oh . Ok. I don't get what Todd is saying then.

This is a prime example why maths and physics are "separated by common language" ;-)

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