The Germination of the Infinitesimal Order (Part 2)

In the room.

Looking at Little Ox's vexed expression, Xu Yun couldn't help but feel a sense of emotion:

Although his character was truly lacking, his mind was absolutely brilliant!

Look at the topics he brought up:

Not to mention Calculus, he even mentioned the concept of normal vectors, the concept of potential energy, the concept of net torque, and the assumption of small deformation.

Each and every one of these concepts was officially published theoretically no earlier than 1807.

What kind of thinking span, from 150 to 200 years, could anyone achieve?

Indeed.

The problem Hooke raised was actually very simple, so simple that Xu Yun immediately thought of nearly twenty solutions. The quickest method was to set up a non-Descartes coordinate system and apply a covariant derivative to solve it.

But don't forget, Xu Yun's knowledge was acquired from later generations. By then, the foundational theories had already been quite thoroughly summarized.

It's like living in an era of controlled nuclear fusion; you could whip up a 200cc engine with your eyes closed.

But what about Little Ox?

He was in the era of rubbing sticks to make fire, yet his gaze reached the calculation formula for the octane number of an internal combustion engine!

Thinking of this, Xu Yun inexplicably felt like laughing:

He had once written a novel, and not only Newton but even Maxwell were criticized by some commenters as "just looked it up, it's just a set of equations."

Afterward, he took a deep breath and turned his focus back to the present:

"Mr. Newton, I highly approve of your line of thinking, but it requires quite a few unknown mathematical tools. The current progress of the mathematical community seems a bit strained."

Little Ox nodded, generously admitting this:

"That's right, but other than that, we would have to use what you called the Han Li Expansion."

After speaking, Little Ox lowered his head again and quickly wrote another line of equations:

V(r)=V(re)+V'(re)(r-e)+[V''(re)/2!](r-re)^2+[V'''(re)/3!](r-re)^3

Then, Little Ox drew a line under this formula and frowned:

"If we use the Han Li Expansion, what would be the properties of the ball near its stable position? This should be a series, but classifying it is also a problem."

Xu Yun looked up at him and said, "Mr. Newton, what if we treat the stable position as a minimum value for calculation?

Let's assume there's a mathematical approach of approximation, meaning… approaching infinity close to 0?"

"Approaching infinity close to 0?"

For some reason, a strange emotion suddenly welled up in Little Ox's heart, like seeing Lisa walking out of the bedroom arm-in-arm with someone else.

But he quickly cast the emotion aside and pondered, "Isn't that the principle of the method of exhaustion?"

The method of exhaustion, which is an early approach to calculating pi, is something anyone who attended elementary school should know.

It actually implies a certain idea:

Even though there's a difference between two quantities, as long as that difference can be made infinitely small, the two quantities can be considered equal in the end.

The method of exhaustion was already considered an abandoned mathematical tool in this era. With Xu Yun's mathematical prowess, being able to casually mention the Han Li Expansion, he theoretically shouldn't make such a mistake of regressing in thought.

Facing Little Ox's question, Xu Yun gently shook his head and said, "Mr. Newton, the concept you mentioned is a non-series variable. But what if we take it a step further and understand it as a series variable?

Even further, what if we consider it a constant that transcends the framework of real numbers?"

"Approaching 0, series variable? Constant?"

Hearing Xu Yun's words, Little Ox was completely stunned.

The concept of infinitesimals was a problem that had caused countless university students to hang themselves from trees.

Generally speaking.

A person's understanding of infinitesimals goes through three stages from undergraduate to doctoral studies.

In the first and second stages, infinitesimals are variables. By the third stage, all infinitesimals become constants, and each infinitesimal corresponds to a specific constant.

These constants do not exist within the framework of real numbers; they are generated by the axioms of non-standard analysis models.

The first stage is the understanding gained while studying mathematical analysis or advanced mathematics in college, where an infinitesimal is as small as you want it to be.

That is, the absolute value of a positive or negative infinitesimal is less than any given positive real number.

The second stage is when studying non-standard analysis, where many calculus formulas introduce infinitesimal quantities, and concepts like "order" appear.

The third stage is when understanding mathematical model theory, at which point infinitesimals can become constants.

Once infinitesimals are understood as constants, a broader mathematical world is revealed, one that is vaster, deeper, and more complex than the currently known mathematical world. Concepts like the second type of limit and their geometric structures emerge. The second type of limit is endowed by an infinite space, while the limit concept of standard analysis is endowed by an infinitesimal space.

This then leads to the compatibility of Euclidean and non-Euclidean geometries, where even intersection points of parallel lines can be accurately represented.

The above situation gave rise to many unconventional geometries, neither Euclidean nor non-Euclidean, but a third type (Chinese geometry), and so on.

What is the practical significance of the understanding of infinitesimals in the third stage?

Most directly speaking, you can go and build supercomputers.

Currently, the most in-depth research on the third stage in China is at USTC. Academician Pan Jianwei and Professor Lu Chaoyang's quantum computers are one of the intuitive manifestations of this.

Friends who have participated in supercomputer algorithm development interviews should all know that the third-order understanding of infinitesimals is a mandatory interview question.

At this time, Little Ox's theoretical knowledge was not yet complete, but as the proposer and founder of Calculus—especially the concept of infinitesimals—he could vaguely provide feedback on this information.

Then Xu Yun picked up his pen and continued to write:

Assuming the coefficient of the linear term is zero at the equilibrium position, then the approximation can only be kept up to the quadratic term, naturally resulting in the form where potential energy is quadratically related to the deviation from equilibrium:

V(r)≈[V''(re)/2!](r-re)^2

V(r)≈k/2(r-re)^2.

Writing up to this point.

Xu Yun stopped writing and glanced at Little Ox, who seemed lost in thought, and quietly turned to leave.

Before leaving, he took a small bag of white sugar, a little salt, half a spoonful of butter, an unused crucible, and two potatoes from the table—the former were common seasonings for breakfast and dinner, while the latter were emergency rations.

Then he tiptoed and gently closed the door.

Little Ox showed no reaction to this; he just stared blankly at Xu Yun's formula, especially the approximate equal sign.

After a few minutes.

His Adam's apple suddenly moved up and down a few times, and a few gurgling sounds came from his mouth.

A moment later, he sprang back to his seat and quickly started writing.

Three hours later.

With a loud crash, Little Ox burst out of the door.

Well, "burst out of the door" in the physical sense—he knocked the door down and carried it out in his hands.

There was no other way; the house was simply too old.

It was past eight in the evening, so Little Ox's first sight was a cluster of firelight in the distance, and Xu Yun's face illuminated by the firelight.

Little Ox quickly walked over to him and said excitedly:

"Fat Fish, I figured it out! It's a force that changes linearly with distance, an elastic force!

Its specific form has no requirements, in other words, any system near a steady state will exhibit elastic behavior!

This is a formula that no one has discovered, a theorem under steady state. I dare bet that Hooke himself didn't derive it, because his given function actually has a zeroth-order term!"

Little Ox mumbled to Xu Yun as he ran. When he reached the campfire, he realized that Xu Yun was currently lowering his head, tinkering with something with a grunt:

"Fat Fish, what are you…?"

"Mr. Newton, you've come at just the right time."

Looking at Little Ox in front of him, Xu Yun picked up a dinner plate and smiled brightly:

"Freshly baked roasted potatoes, delicious with sauce."

"Sauce? What sauce?"

"Ketchup."