The Lord Demon King is Unfathomable!

Chapter 1277 - 377: 178th Void Realm

"Functional?"

Ilana softly repeated this unfamiliar term, her clear eyes filled with curiosity and a quest for the unknown.

"That's right."

Luo Yan's voice was gentle and clear. To help his genius apprentice better understand the mysteries of the functional, he used an easy-to-grasp analogy.

"Ilana, do you remember what we learned about 'Calculus'? What is its core purpose?"

Ilana answered without hesitation.

"It's to study the rate of change of functions and to find the extreme values of functions. For instance, finding the highest point of a mountain peak or the lowest point of a valley."

"Exactly right." Luo Yan nodded approvingly. "Calculus tells you at which 'point' you can stand the highest or lowest. But a 'functional' addresses a completely different issue."

He took out his magic wand and gently waved it, drawing a gentle curve in the air, followed by a steep one.

Through his mastery of Magic Power, without relying on any spell, he could precisely control each flickering "Light Element," moving along the trajectory of his wand.

"Our 'functional' doesn't concern itself with which point is the highest. Rather, if we want to go from point A at the foot of the mountain to point B, with countless paths to choose from, which 'path' is the shortest distance? Which 'path' requires the least effort to traverse?"

"Or, when a flickering light orb slides from point A on the rim of a bowl to point B, which 'curve' does it follow to minimize the time taken?"

Seeing Ilana's eyes suddenly brighten, Luo Yan continued, "You see, what we're seeking is no longer an optimal 'point,' but an optimal 'line'—in other words, a function itself that minimizes the 'total distance,' 'total time,' or 'total expenditure' as a whole."

These words, like a key, opened the tightly locked door in Ilana's mind, instantly unraveling the chaotic threads of her thoughts.

"I understand... So, we're no longer looking for a numerical solution, but... a functional solution! A rule that describes the concept of 'optimal' itself!"

A rare excitement appeared on that cool face—it was the expression of a sailor who, after months of navigating the vast endless ocean, finally sights the coastline!

She paused for a moment there, trying to control her lightly trembling voice before continuing.

"What we're solving for... is a system!"

"Exactly! You've summed it up very well."

Luo Yan smiled with satisfaction, then immediately shifted his tone to continue speaking.

"The dilemma faced by Mr. James Walli, I think, is precisely this problem... I'm very glad you managed to summarize his problem."

He pointed to the notebook, smiling as he continued.

"This gentleman ingeniously hypothesized that when an object moves from one point to another, it chooses a path that minimizes some form of 'action.' But he lacked a vital mathematical tool, one that could sift through these infinite possibilities to find that singular 'optimal path.'

As he observed Ilana remaining in her state of epiphany, Luo Yan's voice tinged with a guiding tone, nudging her through the proverbial door.

"...And that tool is the variational method based on 'functional'!"

"It helps us build a completely new equation, which I call the 'Path Optimization Equation.' So long as every point on a path satisfies the balance conditions described by this equation, then this path is the 'optimal solution' we are looking for."

"Do you know this equation?" Ilana blinked, her face full of admiration.

To her, it seemed like he knew everything.

The very problem she struggled to solve for several days was comprehensively explained by her esteemed Master in just a few words.

Luo Yan intended to say he indeed knew what it was.

Wasn't it the Euler-Lagrange equation? He didn't even need to ask online to know all about it, using it like second nature.

However, to do so would certainly stifle a promising Soul.

The people of this world originally had the opportunity to walk this necessary path through their strength alone, without relying on the intelligence from the other side of the Void.

This was true for civilization, as well as for individuals.

As someone who graduated with outstanding results even in Hell, Luo Yan knew very well that pointing Ilana in the right direction was far more meaningful than force-feeding knowledge.

Therefore, he told a lie and made a different choice than Linte Isaac did back then.

"I, of course, don't know, but I believe with your talent, you will certainly be able to discover its existence on your own."

Talent...

Ilana's face flushed a bit.

The word talent is usually used to describe the potential to comprehend Extraordinary Power, and thus is rarely used to describe ordinary people outside the Nobles.

"Do I... really have such ability?"

"Of course you do. Haven't you already walked further along the path Mr. James Walli took, reaching heights he didn't? I think, his Extraordinary Power was surely above yours, but that doesn't prevent you from 'standing on the shoulders of a Giant' just the same."

Luo Yan smiled gently and, looking into those clear, transparent eyes, continued speaking in a warm voice.

"Now, let me teach you some fundamental techniques of the variational method... Eh, think of it as helping me prepare for my lesson; tomorrow, I will be lecturing on it in the Science class."