You Call This Easily Obtained?

The breeze was lovely today.

A gentle wind swept across the lawn, carrying away the clouds in the sky.

Warm sunlight shone on the blackboard, drawing the attention of all the teachers and students.

Some studied it with folded arms, some looked disdainful, and some watched attentively.

Some were mysterious, some were silent, were you really listening to the story? ()

Li Mo stood with one hand in his pocket. He wasn't trying to look cool, but there was no denying how cool he looked.

He gently wrote the first equation on the blackboard:

[dy/dx=lim(Δx→0)[f(x+Δx)-f(x)]/Δx]

"Some call this expression a fluxion, while others call it a differential, but over a century ago, the archbishop raised a question: is this Δx in the denominator actually zero or not..."

Mr. Harrison suddenly had a premonition that Li Mo had done more than simply come prepared today...

Beside him sat three excellent teachers:

Mr. Walter, who taught applied mechanics, Mr. Vance, who taught algebra, and Collins, a classical geometry teacher known for being particular.

The three teachers reacted differently to the question Li Mo had raised, but their reactions shared the same underlying sentiment:

What exactly is this kid trying to say?

Li Mo continued, "If it's zero, what gives us the right to divide by zero in the first step of the calculation? If it isn't zero, what gives us the right to treat it as zero and discard it in the final step of the derivation?"

Matthew raised his hand and stood up. "Isn't there no point discussing this? The answer is always correct in actual calculations!"

He was very active today. He had to seize every opportunity to show off.

What if this is the chance to show off that the organization prepared for me?!

"Matthew is right. The answer is correct, but I don't think arriving at the correct answer this way is elegant. Today, I'll use purely algebraic inequalities to give limits a precise definition."

Li Mo turned around and raised his right hand.

"Suppose the function f(x) is defined in some deleted neighborhood of the point x₀. If there exists a constant A such that, for any given positive number ε..."

"...no matter how small ε is, there exists a positive number δ such that, when x satisfies the inequality 0<|x-x₀|<δ, the corresponding function value satisfies the inequality |f(x)-A|<ε."

"Then the constant A is called the limit of the function f(x) as x approaches x₀."

The complete mathematical expression was as follows:

[For any ε>0, there exists δ>0 such that, whenever 0<|x-x₀|<δ, |f(x)-A|<ε always holds]

After he finished writing.

Most of the students looked bewildered.

"What is that supposed to be?"

"I don't understand... how do you calculate a limit with that?"

"There isn't a tangent or a velocity curve. What exactly is it expressing?"

Mr. Harrison was rapidly writing it down from memory in his notebook and thinking it over.

The other three teachers grew more attentive as well, trying to follow Li Mo's reasoning.

Li Mo didn't think he was going too fast. He had already given everyone dozens of seconds to think.

After a moment's thought, he considerately added another explanation:

"Everyone, ε is the error tolerance given by the challenger. The challenger can demand that the difference between the function value and A be less than one thousandth, and the prover's task is to provide an interval radius δ. As long as values are taken within that radius, the error will never exceed the challenger's requirement. Static inequalities replace ambiguity. This is the rigor modern mathematics should have."

When Li Mo finished, he figured the teachers should surely be able to understand it now.

So he continued with the derivation:

"Next, we'll prove that a limit is locally bounded. Given lim(x→x₀)f(x)=A, we take ε=1, so there exists a corresponding δ>0..."

Breaking down the absolute-value inequality: [|f(x)|=|f(x)-A+A|≤|f(x)-A|+|A|<1+|A|]

After writing it down, Li Mo said:

"From this, using the triangle inequality, we easily obtain that the function f(x) is locally bounded in that deleted neighborhood."

He promptly drew a horizontal line.

He prepared to move on to the next section, proving the uniqueness of a limit.

The people below were completely lost now.

Teacher Vance, the only one who had barely been able to keep up, was now staring blankly at the blackboard as well.

How was that easily obtained? Where did "easily obtained" come from?

How on earth did the absolute-value expression jump to 1+|A|?

One top math student couldn't hold back either. He raised his right hand and said hesitantly, "Sorry... Li Mo, please wait a moment!"

Li Mo stopped. "Do you have a question?"

The top student stood up and asked with difficulty, "You just said... easily obtained? How exactly did you get that? I don't quite understand..."

Li Mo froze for a moment.

This was an absolute-value bound from introductory analysis, something he would have learned in his first year of university in his previous life.

For him, it really was easily obtained.

But in this era, it was clearly beyond the syllabus.

Li Mo nodded gently. "Oh, sorry. I skipped three steps in bounding the absolute value here. I'll go through them again in detail."

He picked up the blackboard eraser and wiped away the abbreviated line.

"Let's go back a step. Using the triangle inequality for absolute values, for any real numbers a and b, |a+b|≤|a|+|b| always holds. Here, we let a=f(x)-A and b=A..."

He broke it down in detail, step by step.

"If we consider only the deleted neighborhood, the upper bound is 1+|A|. If the function is also defined at the center point x₀ itself, we include it as well, and the bound over the entire closed neighborhood is M=max{|f(x₀)|,1+|A|}."

"Brilliant... brilliant!"

In the silence, Mr. Harrison couldn't help standing up and exclaiming!

The students looked at one another.

Most of them hadn't understood.

But the teacher had stood up and called it brilliant... so it was probably correct, right?

So someone began to clap in confusion.

As soon as one person started, the others hurried to follow.

A burst of applause followed.

Matthew's expression wasn't looking too good.

Wasn't I supposed to get to show off? How did Li Mo end up showing off again?

Was this Mystical Ritual given to the wrong person?!

Seeing everyone applaud, Li Mo figured they must have understood, so he naturally continued:

"Let's look at a practical problem."

"This is a problem I ran into recently while helping improve the steam engine at my family's spinning mill."

Steam engines in this era had a centrifugal flyball governor.

Two metal balls hung on either side of a rotating shaft, connected to it by linkages.

As the shaft's rotational speed increased.

Centrifugal force drove the metal balls outward, which in turn controlled how far the steam inlet valve opened through a lever linkage.

"In the steam engines at Mendel Port and the major textile mills, workers often face the problem of governor flyball oscillation. During actual operation, the governor often vibrates, sometimes even causing steam pipes to burst, right?"

Several students from the mechanics department perked up at this.

This was a malfunction they often encountered during their practical training!

"Why does this happen? Because this should be a dynamical system with multiple degrees of freedom."

Li Mo wrote down a closed-loop perturbation characteristic equation:

[d³(Δθ)/dt³+a₁²(Δθ)/dt²+a₂(Δθ)/dt+a₃·(Δθ)=0]

"Let the perturbation in the flyball opening be Δθ, and let the perturbation in the main shaft's angular velocity be linearized around the operating point. If we want to completely eliminate the flyballs' self-excited vibration and have the system asymptotically converge back to equilibrium after a sudden change in the spinning machine's load..."

"And ensure that the maximum dynamic overshoot of the main shaft's rotational speed during the decay process does not exceed 5%..."

"Based on this cubic algebraic characteristic equation, please give the necessary and sufficient conditions for the real parts of all the system's characteristic roots to be negative, and use them to calculate which critical threshold the mechanical damper coefficient among these coefficients must exceed."

After writing out the problem, Li Mo looked at the people below:

"It's a fairly simple control problem. Well, does anyone want to come up and solve it?"