Inside the room.
Looking at Little Niu's frustrated face, Xu Yun couldn't help but feel a surge of emotion:
Although his character is really lackluster, his mind is truly exceptional!
Just look at what he's mentioned:
Not only calculus, but also concepts of normal vectors, potential energy, net torque, and the hypothesis of small deformations.
Each of these concepts, when made formal, couldn't have been publicly outlined until after 1807 at the earliest.
This kind of 150 to 200-year mental leap... who could achieve it?
Indeed.
The problem Hook proposed is really simple, so simple that the first solution Xu Yun thought of came up with was nearly twenty methods, the quickest method would solve it by setting up a non-Cartesian coordinate system and a covariant derivative.
But don't forget, Xu Yun's knowledge was gained through later learning, when the fundamental theories were already quite well rounded.
Like mastering controlled nuclear fusion, it's like making a 200cc engine with eyes closed.
But as for Little Niu?
He belongs to the era of making fire by friction, yet his vision has already seen the outrageous cetane number calculation of an internal combustion engine!
Thinking of this, Xu Yun felt a strange urge to laugh:
He once wrote a novel, and never mind Newton, even Maxwell was dissed by some comments as 'just a set of equations'.
Then he took a deep breath and refocused on the scene:
"Mr. Newton, I really agree with your thinking, but it requires too many unknown mathematical tools, and with the current progress in mathematical research, it seems a bit lacking..."
Little Niu nodded, readily admitting this point:
"That's right, but besides this, you must have used what Han Li expanded on."
After saying this, Little Niu lowered his head again and quickly wrote another line of formula:
V(r)=V(re)+V'(re)(r-e)+[V''(re)/2!](r-re)^2+[V'''(re)/3!](r-re)^3...
Next, Little Niu drew a line under this row of formulas and frowned, saying:
"If Han Li's expansion is used, what should be the property of an elastic ball near a stable position? It should be a series, but splitting it into parts is indeed a problem."
Xu Yun looked up at him and said:
"Mr. Newton, what if we calculate the stable position as a local minimum?
We can assume a mathematical approximation, which means... infinitely approaching zero?"
"Infinitely approaching zero?"
For some reason, a strange emotion suddenly surged in Little Niu's heart, like seeing Lisa come out of the bedroom hand-in-hand with someone else.
However, he quickly cast aside this emotion and pondered for a moment, saying:
"Isn't that the idea of the circle quadrature method?"
The circle quadrature method, that is, an early idea for calculating pi, which anyone who attended elementary school should know this method.
It actually suggests this kind of idea:
Although there's a gap between two quantities, as long as this gap can be infinitely reduced, it can be considered that eventually, the two quantities will be equal.
In this era, the circle quadrature method is already considered an abandoned mathematical tool; with Xu Yun's mathematical expertise of casually mentioning Han Li's expansion, he theoretically shouldn't make such retrogressive errors in thinking.
Facing Little Niu's doubts, Xu Yun gently shook his head and said:
"Mr. Newton, the concept you are mentioning is a non-series variable, but what if we go a step further and understand it as a series variable?
Even further, regarding it as a constant beyond the real number framework?"
"Approaching zero, series variable? Constant?"
Hearing Xu Yun's words, Little Niu was suddenly stunned.
The concept of infinitesimals is a problem that has hung many university slackers on the tree.
Generally speaking.
A person from being a university student to a Ph.D. has to go through three stages of understanding infinitesimals.
The infinitesimals of the first and second stages are variables, and upon reaching the third stage, all infinitesimals become constants, with each infinitesimal corresponding to a constant.
These constants are not within the real number framework; they are generated by the axioms of the non-standard analysis model.
The first stage is the cognition during college when learning mathematical analysis or advanced mathematics, where infinitesimals can be as small as possible.
That is the absolute value of positive and negative infinitesimals, less than any given positive real number.
The second stage is during learning non-standard analysis when many calculus formulas introduce infinitesimals, and concepts like sequences appear.
The third stage is recognizing mathematical model theory, where infinitesimals can become constants.
Once one realizes infinitesimals are constants, one discovers a broader mathematical world; this mathematical world is broader, deeper, and more complex than the known mathematical world today, with the second type of limit idea and its geometric structure emerging. The second type of limit idea is endowed by the infinite large space, and the standard analysis's limit idea is endowed by the infinitesimally small space.
Next came the phenomenon of compatibility between Euclidean and non-Euclidean geometries, where parallel intersection coordinates can be accurately represented.
The above situation also gave rise to many unconventional geometries, which are neither Euclidean nor non-Euclidean, belonging to a third type of geometry (Chinese geometry), etc.
What practical significance does the third stage of understanding infinitesimals have?
Very directly, you can engage in supercomputers.
Currently, the most in-depth research in this third stage domestically is USTC, with Pan Jianwei academician and Lu Chaoyang professor's quantum computer being an intuitive manifestation of it.
Friends who have participated in the supercomputer algorithm development interview should know that the third-order understanding of infinitesimals is a mandatory exam question in the interview.
Although Little Niu's theoretical knowledge wasn't so well-rounded at this time, as the proposer and foundational figure of calculus—especially infinitesimals, he could vaguely respond to this information.
Then Xu Yun took a pen and continued writing:
Assuming the coefficient of the first-order term is zero at the point of equilibrium, then it can only preserve up to second-order approximation, naturally obtaining the form of potential energy quadratic with the deviation from equilibrium:
V(r)≈[V''(re)/2!](r-re)^2
V(r)≈k/2(r-re)^2.
After writing here.
Xu Yun put down his pen, glanced at the somewhat dazed Little Niu, and quietly turned to leave.
Before leaving, he picked up a small bag of sugar, a bit of salt, half a spoon of butter, an unused crucible, and two potatoes from the table—earlier ones being commonly used seasonings for breakfast and dinner, the latter being emergency reserve food.
Then he tiptoed and gently closed the door.
Little Niu showed no reaction to this; he just stared dazedly at Xu Yun's equation, especially the approximately equal sign.
A few minutes passed.
His Adam's apple suddenly moved up and down a few times, and a few gurgling noises emitted from his mouth.
Moments later, he dashed back to his seat, quickly picking up a pen.
Three hours later.
With a bang, Little Niu rushed out the door.
Yes, physically rushed out the door—he knocked the door off, holding it in his hands.
There's no helping it, the house was just too old.
It was just over eight in the evening, so Little Niu's first sight was a cluster of firelight not far away, with Xu Yun's face reflected in the light.
Little Niu quickly walked to his side, excitedly saying:
"Fat Fish, I figured it out, it's a linearly changing force with distance, an elastic force!
Its specific form has no particular requirements, in other words, any system near the steady-state, will exhibit elastic behavior!
This is a formula that hasn't been discovered by someone, a theorem under steady-state, I bet, Hook himself didn't derive it because his function actually has a zeroth-order term!"
Little Niu mumbled as he ran to Xu Yun, only to realize when he got to the fire that Xu Yun was actually nodding his head, fiddling with something:
"Fat Fish, what are you...?"
"Mr. Newton, you came just in time."
Gazing at Little Niu in front of him, Xu Yun picked up a plate and smiled brilliantly:
"Freshly baked potatoes, delicious with sauce."
"Sauce? What sauce?"
"Tomato sauce."
...
Note:
Remember the tomato mentioned when introducing the utensils earlier, heh heh...