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Inside the room.

Looking at little Newton's frustrated face, Xu Yun couldn't help but feel emotional:

Although this guy's character is really lacking, his brain is truly amazing!

Let's take a look at what he mentioned:

Calculus aside, he also mentioned the concepts of normal vectors, potential energy, net torque, and the assumption of small deformations.

Each of these concepts is a separate one, officially introduced in theory after 1807.

This kind of thinking span of 150 to 200 years... who could achieve that?

Indeed.

The problem Hooke proposed is actually quite simple. It was so simple that Xu Yun came up with nearly twenty solutions in his mind, and the quickest method was just to calculate a covariant derivative on a non-Cartesian coordinate system.

But don't forget, Xu Yun's knowledge was acquired through studying in the future, where the foundational theories had already been well-established.

It's like living in an era where controlled nuclear fusion is mastered, and even with your eyes closed, you can create a 200cc engine.

But what about little Newton?

He belongs to an era where they're still rubbing sticks together to make fire, yet he somehow managed to come up with the formula for octane rating of internal combustion engines!

Thinking of this, Xu Yun couldn't help but smile:

He once wrote a novel, and as a result, not only Newton, but even Maxwell was dissed by some comments as "I looked it up, it's just a system of equations."

Then he took a deep breath and refocused on the present:

"Mr. Newton, I really appreciate your line of thinking, but it requires quite a few unknown mathematical tools. It seems that the current progress in the mathematical field is a bit lacking..."

Little Newton nodded and generously admitted this:

"That's right, but besides that, we must also use the Han Li expansion you mentioned."

After saying that, little Newton lowered his head again and quickly wrote down a line of equations:

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

Then he drew a line under this line of equations and furrowed his brow:

"If we use the Han Li expansion, what would be the properties of the ball near its stable position? It should be a series, but dividing it up is a problem."

Xu Yun glanced at him and said:

"Mr. Newton, if we treat the stable position as a local minimum for calculation?

Let's assume a mathematical approximation, which is... infinitely approaching zero?"

"Infinitely approaching zero?"

For some reason, a strange emotion suddenly arose in little Newton's heart, as if he had seen Lisa walking out of the bedroom holding hands with someone else.

But he quickly dismissed this emotion from his mind and pondered for a moment:

"Isn't that the method of exhaustion?"

The method of exhaustion, which is actually an early approach to calculating pi. Those who have gone through primary school should know about this method.

It implies the following idea:

Although two quantities may have a difference, as long as this difference can be infinitely reduced, it can be considered that the two quantities will eventually be equal.

In this era, the method of exhaustion is already considered an abandoned mathematical tool. It's theoretically impossible for Xu Yun, who can casually mention the Han Li expansion, to make such a regression in thinking.

Facing little Newton's question, Xu Yun gently shook his head and said:

"Mr. Newton, the concept you mentioned is a non-series variable, but if we take it a step further, what if we understand it as a series variable?

Or even further, as a... constant beyond the framework of real numbers?"

"Approaching zero, series variable? Constant?"

Upon hearing Xu Yun's words, little Newton was completely stunned.

The concept of infinitesimals, it's a question that has stumped countless university students.

Generally speaking.

From being a university student to becoming a doctorate, one's understanding of infinitesimals goes through three stages.

In the first stage and the second stage, infinitesimals are considered variables. When one reaches the third stage, all infinitesimals become constants, and each infinitesimal corresponds to a constant.

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

The first stage is the understanding during the study of mathematical analysis or advanced mathematics in university. At this stage, infinitesimals are variables, meaning they can be as small as possible.

That is, the absolute value of positive and negative infinitesimals is smaller than any given positive real number.

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

The third stage is when one understands mathematical model theory. At this stage, can infinitesimal quantities become constants?

Once one realizes that infinitesimal quantities can be constants, they will discover a broader mathematical world. This mathematical world is wider, deeper, and more complex than the currently known mathematical world. The second type of limit thinking and its geometric structure emerge. The second type of limit thinking is endowed by the infinite space, while the limit thinking of standard analysis is endowed by the infinitesimal space.

Then the compatibility phenomenon between Euclidean geometry and non-Euclidean geometry appears, and the coordinates of parallel intersections can be accurately represented.

These situations have given rise to many unconventional geometries, which are neither Euclidean nor non-Euclidean, but belong to a third type of geometry (Chinese geometry), and so on.

What practical significance does the understanding of infinitesimals in the third stage have?

The most direct answer is that you can go and work on supercomputers.Currently, the most in-depth research on the third stage in China is being conducted by the University of Science and Technology of China. The quantum computer developed by Academician Pan Jianwei and Professor Lu Chaoyang is one of the intuitive manifestations in this field.

Those who have participated in the interview for the development of supercomputer algorithms should know that infinitesimal third-order cognition is a must-ask question in the interview.

At this point, although Little Newton's theoretical knowledge is not so perfect, as the proposer and founder of calculus - especially the concept of infinitesimals, he can vaguely respond to this information.

Then Xu Yun took the pen and continued to write:

If the first-order coefficient of the association is zero at the equilibrium position, then it can only be retained to the second approximation at the minimum, and naturally, a form related to the potential energy and the deviation from the equilibrium is obtained.

V(r) ≈ [V''(re)/2!](r-re)^2

V(r) ≈ k/2(r-re)^2.

Having written this far,

Xu Yun put down his pen, glanced at the somewhat dazed Little Newton, and quietly turned and left.

Before leaving, he took a small packet of sugar, a bit of salt, half a spoonful of butter, a disused crucible, and two potatoes from the table - the former are common condiments for breakfast and dinner, and the latter are emergency food reserves.

Then, on tiptoe, he gently closed the door.

Little Newton was completely oblivious to this, he just stared blankly at Xu Yun's formula, especially the approximate equal sign.

A few minutes later,

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

A moment later, he darted back to his seat and quickly picked up his pen.

Three hours later,

With a bang, Little Newton burst out of the door.

Well, literally burst out of the door - he knocked the door down and carried it in his hand.

There was no choice, the house was just too old.

At this time, it was just after eight o'clock in the evening, so the first thing Little Newton saw was a cluster of fire not far away, and Xu Yun's face illuminated by the fire.

Little Newton quickly walked to his side, excitedly said:

"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 show elastic behavior!

This is a formula that no one has discovered, a theorem under steady state, I bet, even Hooke himself didn't derive it, because the function he gave actually has a zero-order term!"

As Little Newton ran and mumbled to Xu Yun, he didn't notice until he reached the fire that Xu Yun was fiddling with something:

"Fat Fish, what are you...?"

"Mr. Newton, you're just in time."

Looking at Little Newton in front of him, Xu Yun picked up a dinner plate, smiling brightly:

"Freshly baked potatoes, they're delicious with sauce."

"Sauce? What sauce?"

"Tomato sauce."

......

Note:

Remember the tomatoes mentioned when introducing the tableware earlier, eh hehe....