Inside the room.
Looking at Little Ox's rueful expression, Xu Yun couldn't help feeling deeply moved:
The man might have been a complete mess in terms of character, but his mind was absolutely brilliant!
Just look at what he'd brought up:
Never mind calculus—he'd also mentioned the concept of normal vectors, potential energy, net torque, and the assumption of small deformations.
Every single one of those concepts had first been formally published as a theory no earlier than 1807.
Who else could span a gap of 150 to 200 years in their thinking?
To be fair.
The problem Hooke had raised was actually very simple—so simple that Xu Yun had thought of nearly twenty solutions right away. The quickest would have been to set up a non-Cartesian coordinate system and apply a covariant derivative.
But Xu Yun had to remember that he'd gained his knowledge through later education, by which time fundamental theories had already been thoroughly systematized.
It was like living in an age with Controllable Nuclear Fusion and being able to build a 200cc engine with your eyes closed.
But Little Ox?
He was living in an age when people started fires by rubbing sticks together, yet somehow his gaze had landed on the cetane number formula for an internal combustion engine. It was that absurd!
Thinking of this, Xu Yun felt an inexplicable urge to laugh:
He'd once written a novel, and the comments had mocked even Maxwell, let alone Newton, as "I looked it up, and it's just a system of equations."
Then he took a deep breath and turned his attention back to the scene:
"Mr. Newton, I agree with your approach, but it requires too many mathematical tools we don't yet have. Given the current state of mathematical research, it seems a bit beyond our capabilities."
Little Ox nodded, openly admitting as much:
"That's true. But otherwise, we'd have to use that Han Li Expansion you mentioned."
After speaking, Little Ox lowered his head again and quickly wrote down 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 beneath the formula and said with a frown:
"If we use the Han Li Expansion, what properties would the ball have near its stable position? It should be a series, but how to divide it up is another problem."
Xu Yun looked up at him and said:
"Mr. Newton, what if we calculate the stable position as a minimum?
Let's assume a mathematical approach—that is, something infinitely close to zero?"
"Infinitely close to zero?"
For some reason, a strange feeling suddenly welled up inside Little Ox, like seeing Lisa come out of a bedroom holding hands with someone else.
But he soon put the feeling aside and thought it over before saying:
"Isn't that just the principle behind the method of exhaustion?"
The method of exhaustion was an early way of calculating pi, and anyone who'd been to elementary school should know about it.
It implied the following idea:
Although two quantities differ, if that difference can be made infinitely small, then the two quantities can be considered equal in the end.
By this era, the method of exhaustion had already been discarded as a mathematical tool. Given Xu Yun's mathematical knowledge—he could casually bring up the Han Li Expansion, after all—he theoretically shouldn't have made such a backward leap in his thinking.
Faced with Little Ox's question, Xu Yun gently shook his head and said:
"Mr. Newton, the concept you're describing is a variable, not a series. But what if we take it a step further and understand it as a series variable?
Or take it one step further and see it as a constant that transcends the framework of the real numbers?"
"Approaching zero, a series variable? A constant?"
Hearing Xu Yun's words, Little Ox froze.
Infinitesimals were a problem that had caused countless college students to fail their courses.
Generally speaking,
A person's understanding of infinitesimals went through three stages between college and the doctorate.
In the first and second stages, infinitesimals were both variables. By the third stage, all infinitesimals became constants, each corresponding to a specific constant.
These constants weren't part of the real-number framework. They were generated by the axioms of nonstandard analysis models.
The first stage was the understanding people gained when they studied mathematical analysis or advanced mathematics in college: an infinitesimal could be as small as you wanted.
That is, the absolute value of a positive or negative infinitesimal was less than any given positive real number.
The second stage came when studying nonstandard analysis. Many calculus formulas introduced infinitesimal quantities, along with concepts such as order.
The third stage came with the study of mathematical model theory. At this point, infinitesimals could become constants.
Once you understood infinitesimals as constants, you'd discover a much broader mathematical world—one wider, deeper, and more complex than the mathematical world known at the time. It contained a second kind of limit concept and its geometric structure. The second kind of limit concept came from infinite spaces, while the limits of standard analysis came from infinitesimal spaces.
Then came the compatibility of Euclidean and non-Euclidean geometry, in which even the coordinates of parallel intersection points could be represented precisely.
This, in turn, gave rise to many unconventional geometries. They were neither Euclidean nor non-Euclidean, but belonged to a third kind of geometry—Chinese geometry—and so on.
So what practical significance did this third-stage understanding of infinitesimals have?
The most direct answer was that you could work on supercomputers.
At present, USTC was leading the country in research at the third stage. Academician Pan Jianwei and Professor Lu Chaoyang's quantum computers were one clear manifestation of this field.
Anyone who'd interviewed for a supercomputer algorithm research position knew that third-order understanding of infinitesimals was a guaranteed interview question.
Little Ox's theoretical knowledge wasn't yet that complete, but as the proposer and founder of calculus—especially the concept of infinitesimals—he could vaguely respond to this information.
Xu Yun then picked up the pen and continued writing:
Suppose the coefficient of the first-order term was zero at the equilibrium position. Then the lowest-order approximation we could retain was the quadratic term, naturally giving us a form in which potential energy was proportional to the square of the displacement from equilibrium:
V(r)≈[V''(re)/2!](r-re)^2
V(r)≈k/2(r-re)^2.
At this point,
Xu Yun stopped writing. He glanced at Little Ox, who seemed lost in thought, then quietly turned and left.
Before going out, he took a small packet of sugar, a little salt, half a spoonful of butter, an unused crucible, and two potatoes from the table. The first few were seasonings commonly used for breakfast and dinner, while the last two were emergency provisions.
Then he tiptoed away and gently closed the door.
Little Ox didn't react at all. He just stared blankly at Xu Yun's formula, especially the approximately-equal sign.
A few minutes passed.
His Adam's apple suddenly bobbed a few times, and a couple of gurgling sounds came from his mouth.
A moment later, he dashed back to his seat and quickly started writing.
Three hours later.
With a loud crash, Little Ox burst out through the door.
In the literal sense—he'd knocked the door off its hinges and was carrying it in his hands.
There was no helping it. The house was simply too old.
It was a little past eight in the evening, so the first thing Little Ox saw was a cluster of flames in the distance, and Xu Yun's face illuminated by the firelight.
Little Ox hurried over to him and said excitedly:
"Fat Fish, I've worked it out! It's a force that varies linearly with distance—a restoring force!
Its specific form doesn't matter. In other words, any system near a steady state will exhibit elastic behavior!
This is a formula nobody's discovered before, a theorem for steady states. I'd bet Hooke himself never derived it, because the function he gave even has a zeroth-order term!"
Little Ox shouted at Xu Yun as he ran. It wasn't until he reached the fire that he noticed Xu Yun was bent over, busily tinkering with something:
"Fat Fish, what are you...?"
"Mr. Newton, you've come at just the right time."
Xu Yun picked up a plate and gave Little Ox a bright smile:
"Freshly baked potatoes. They're delicious with sauce."
"Sauce? What kind?"
"Ketchup."
Before you continue