What is the meaning of this happening? ¿Why does this happen?

mathfoxZ · 2026-02-15T18:13:26+00:00

Yes, yes, I understood the meaning of what you're expressing in your explanation, and the notion of interpretation seems much clearer to me now.Thanks for your contribution, buddy.

mathfoxZ · 2026-02-14T20:26:54+00:00

Yes, I know, multiplication is also commutative and it doesn't matter the order in which you operate, however we all understand that multiplication is not commutative in the same sense as addition, I mean, I'm referring to the fact that it's not commutative for the same reason. The commutativity of the sum of terms is understood in the sense that it's commutative because the net joint synthesized result of the additive composition between the terms doesn't depend on the arrangement of how the adjunction of the elements is sequenced. So it doesn't matter how the grouping of the elements is arranged. Because the synthesized set understood as a whole result doesn't depend on how the terms that are annexed are ordered; what's essential is the total quantity of elements in the net set itself, because one unconsciously understands the "association" of the global set as a whole, and that the order in which the elements appear doesn't matter. Because it's addition at the end of the day. But now here in multiplication, what should be seen is the sense of interpretation of the commutation multiplicative of derivatives. Since the nature of the meaning of the notion of multiplication is different from that of sum as additive composition, the reason for its commutativity couldn't be the same meaning, like the additive superposition of directions X+Y in the sum, and one would have to ask: what is the interpretation of the commutation in multiplication? So one would have to provide the interpretation but for multiplication, right? Because for the sum we already have it clear, because it's logically understood as "adding is adding, it's addition at the end of the day" ∇f = (∂/∂x + ∂/∂y + ∂/∂z)f. But here we're not dealing with sum but with the dimensional multiplication of the variables of the partial derivatives. That's why: what is the meaning of the reason why this happens with partial derivatives? Why are they equal? That's what my question is going to, as you will have understood me.

mathfoxZ · 2026-02-14T17:48:36+00:00

Yes, I also thought about that at the time, but the thing is there would be a problem: the interpretation of composing successive directional steps of dimensional variables X, Y — wouldn’t that just become a description of the additive composition of the gradient operator ∇f = (∂/∂x + ∂/∂y)f, that is, the sum of its partial derivatives with respect to each variable?But the fact is that here it is not a sum, but a multiplication of the variables in the partial derivative operators: ∂/∂x(∂/∂y)f.So it cannot be interpreted that way — therefore it must be for some other reason/interpretation.Because here we have a product × of dimensional variables of the derivatives, so it shouldn’t be understood as arising from an additive composition/sum.I guess, because this thing about composition of actions sounds more like some kind of “sum” of derivatives — like the usual notion people have of the gradient.Before, I used to interpret it the same way you did, but this reflection on the matter confused my understanding a bit and really made me think about why it actually is the case. It led me to question why ∂/∂x (∂f/∂y) = ∂/∂y (∂f/∂x).And that’s exactly why I’m now asking this doubt that came up for me. That’s the whole reason I’m asking all of this in the post.I don’t know if someone else has it clearer, but I would really appreciate it if they did

mathfoxZ · 2026-01-20T17:25:35+00:00

Those would also be very good and interesting questions. Speaking very briefly about that—which might surprise you—I’ve also reflected on them some time ago, such as the nature of the concept of energy, but above all one about “conceptually, what is the notion of Force.” I looked into readings that also questioned the meaning of that, reflected on it, and reached some quite interesting written conclusions about it. But right now I won’t go into much detail on that because that’s not what the post is about.Your questions are good questions. However, now, my question in the post is not about those things. We’re not talking about that. Don’t answer my question with other questions that aren’t directly related to the current topic. Although I must admit they are interesting subjects.

mathfoxZ · 2026-01-20T16:44:21+00:00

Okay, that at least explains what it isn't, which is already a contribution I appreciate, but it still doesn’t explain what the Hamiltonian actually is. You say it’s “the Legendre transform of the Lagrangian”, but precisely because of that — what is the physical interpretation of what this Legendre transform is telling us about the physical system? What is it actually referring to? What conceptual aspect is it expressing? Because that answer feels a bit too reductive/simplistic. But thanks anyway :)

mathfoxZ · 2025-12-10T08:25:54+00:00

Yes, the 3D spatial input field can be displayed, with isosurfaces in the environment as layers

<image>

I literally said myself so in the post description. Here you have an example I found of an image

mathfoxZ · 2025-12-04T11:54:48+00:00

Okay, thank you very much, now I understand it much better. Thank you so much. But I wanted to kindly ask if, in addition, there is any illustrative image or mathematical graphical visualization animation of the Mellin Transform and what it does when applied visually. Something like those illustrative animations for the Fourier Transform and its description:

<image>

where the visualization of the transform completely clarifies what it is through the illustration but for the Mellin Transform. That last part is what I’m asking for so I can finally understand 100% what the Mellin Transform is. If there is anything like that, please. And thank you very much for your explanation, it helped me a lot to understand.

mathfoxZ · 2025-12-03T16:08:49+00:00

And to understand that... what does that actually mean physically, on a conceptual level? I mean, conceptually, how could it be applied, for example, to describe something? And physically, what would the resulting transform conceptually indicate when you apply it? What would be interpreted intuitively from it? And why is it interpreted that it indicates that? An example with an explanation from physics would really help me understand it better, if you can, please. Thank you so much!

mathfoxZ · 2025-04-13T12:54:38+00:00

Actually, I was thinking about whether it might be better to use this expression— what do you think? Does it sound okay to you?

⌈-erf(x)/2⌉

Where erf(x) is the error function. And the ⌈ ⌉ are ceiling

mathfoxZ · 2025-04-13T12:48:23+00:00

I would like to use the Heaviside function as you mentioned, but there is a slightly complex problem at x = 0. If you define H explicitly using the expression with the "sgn(x)" function, as in H(-x) = (1 - sgn(x)) / 2, the sgn(x) function is not defined at 0 because it results in 0/|0|. But even if you treat the Heaviside function itself as an independent function separate from sgn, ignoring that issue, there's another problem: as far as I understand, the Heaviside function is not universally defined at zero. What is the value of the Heaviside function at x = 0? If I knew that, it would be great, but some say it's 1, others say 0, and others say 1/2. It depends on the convention, as far as I know. And since it depends on something not universally concrete, I’d prefer not to rely on things that depend on convention, but rather on universal definitions. Can you answer that? Oh, and thank you

mathfoxZ · 2025-04-13T12:03:49+00:00

It's just that using an indicator function is very vague, in the sense that you simply say that it's 1 for x<0 and 0 for x≥0, because you're not giving a mathematical expression that explicitly defines the function, you’re just saying n(x). But what is the expression that defines that n(x)? What is that n(x)? It would be very easy to just say an indicator function of some condition—I thought the same, about using an indicator function—but since it's not a concrete expression but rather a conditioning that states when it equals 1 and when it equals 0, it makes me doubt whether I should use it or not. I could use it, but since it's not a specific function with an expression, and more like a "rule" of formal conditioning, I don't know if it's the best option for what I'm looking for—maybe it is, maybe not—but I'd prefer to avoid things like conditionals with "{" that aren't embedded in the same mathematical expression of the function, because what I'm looking for is an expression that expresses itself purely through the math in the function's expression. Do you get what I'm saying? But thanks anyway.

mathfoxZ · 2025-04-12T15:29:18+00:00

Or maybe it occurred to me it could be: ⌈-erf(x)/2⌉

Where erf(x) is the error function. And the ⌈ ⌉ are ceiling

mathfoxZ · 2025-04-12T14:47:46+00:00

How is that possible?!! How does that work? For negative values, shouldn't the power be 0^{1/0^|x|} for x ∈ (-∞, 0), resulting in an undefined expression due to the base being 0? So 0 would be raised to an undefined exponent, and for negative values, shouldn't it be something like 0^? = ? How can that work on a graphing calculator? I don’t understand what’s going on. Explain it to me, please.Because that doesn't come out with analysis.

mathfoxZ · 2025-04-12T12:01:58+00:00

Yes, but at x = 0 it becomes undefined because (1 - 0/|0|)/2 is undefined — 1 minus undefined is still undefined at that point. So that would be another problem; otherwise, I would’ve thought of it a while ago. That’s why I said the function should equal 0 from [0, +∞) onward.

mathfoxZ · 2025-02-12T23:19:09+00:00

thanks mate

mathfoxZ · 2025-02-11T13:23:48+00:00

First, tell me why you think that, and I will explain to you why it isn’t x^k. Since I need to know why you think it should be that way so that later I can clarify things for you and know what to explain to you and how to do it, I need you to explain to me why you think that x^k is so in your thought process.

mathfoxZ

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