PDLP Chapter 4 This cutting-edge technology is truly amazing.
Chapter 4 This cutting-edge technology is truly amazing.
The library of City University of Hong Kong is an iconic building, with its red brick exterior covered in ivy, which appears serene and solemn under the autumn sun.
Lu Feng went straight up to the second floor, which was the exclusive area for science and engineering books.
When I pushed open the door, there were about ten people sitting sparsely inside, most of them with their heads down, either reading or typing papers on their laptops.
Behind the management desk at the entrance sat a girl wearing glasses, next to an old-fashioned desktop computer with the borrowing registration system running on the screen.
Lu Feng glanced around and saw that there were still empty seats in the row by the window.
He put his schoolbag on the table and went straight to the bookshelf.
He had spent over a decade studying mechanical principles in his previous life, so he didn't need much additional study in the short term.
But advanced mathematics is different; it's the fundamental language of all science and engineering disciplines, and it was also one of his biggest weaknesses in his previous life.
"Mathematics must be improved first."
Lu Feng pulled out a copy of "Advanced Mathematics" (Tongji University edition) from the bookshelf, and also grabbed a matching exercise book, before returning to his seat.
Let's turn to the first chapter, Functions and Limits.
He had learned these things in his previous life and used them sporadically in his work, but he hadn't listened to them carefully during his university years and had to make up for it all through self-study later, so his foundation wasn't solid.
But it's different now.
What was the most difficult thing about learning math in the past?
attention.
After reading for twenty minutes, my hand unconsciously reaches for my phone, and after scrolling through short videos for ten minutes, half an hour has passed before I realize it, and I've forgotten half of what I read.
The cycle repeats itself endlessly, resulting in shockingly low efficiency.
But with efficient learning, Lu Feng was able to completely immerse himself in it.
The continuity of functions, the ε-δ definition of limits, the squeeze theorem... these concepts jumped out of the textbook and corresponded one by one with the scenarios he encountered in his previous life's actual work.
He used to find the definition of ε-δ so boring that he wanted to die, but now he suddenly understands it.
"Isn't this just the mathematical abstraction of tolerance control in precision machining?"
"Give me an error range ε, and I can find a processing precision δ to ensure that the product falls within the acceptable range."
The practical experience from my past life and the theoretical knowledge I'm learning again in this life seem to have created a chemical reaction.
Another hour has passed.
Derivatives, differentials, mean value theorem, Taylor expansions... these concepts are digested at an unreal speed under efficient learning conditions.
When he closed the last page of the chapter on differential equations, the sunlight outside the window had changed from slanting to direct.
Lu Feng glanced at his phone; it was 11:48 PM.
In two hours, we derived the equations from the limits of functions all the way to the differential equations.
He leaned back in his chair and rubbed his slightly sore eyes.
"At my previous learning pace, this amount of material would have been enough for me to study for a week."
"This passive skill of the system is even more outrageous than I imagined."
Before getting up to eat, Lu Feng glanced one last time at the differential equation concerning the volume of solids of revolution, the last exercise he had just completed.
"Using the portion of the curve y = sin(x) within the interval [0, π] as its contour, rotate it around the x-axis for one revolution, and calculate the volume of the resulting solid of revolution..."
"A simple idea of the infinitesimal method came to mind: construct a thin disk with a thickness of dx, a radius of sin(x), and an area of π[sin(x)]², and then integrate from 0 to π."
After figuring this out, Lu Feng tucked the draft paper into his textbook and silently uttered "System" in his mind.
A light blue panel popped up.
Current learning value: 300
Lu Feng's eyes lit up.
"Two hours, 300 learning points."
"At this rate, if I study for another afternoon, I can get 1000 learning points, which is just enough to exchange for a black technology blueprint."
"So awesome!"
Lu Feng put away the panel, stacked the books on the table to reserve a spot, and got up to go downstairs to the cafeteria.
The cafeteria is located on the south side of the teaching area, about a seven or eight-minute walk away.
Lu Feng casually ordered a plate of scrambled eggs with tomatoes, beef stew with potatoes, and a bowl of rice. He found a corner to sit down and finished it off in a few bites.
His mind didn't stop even while he was eating; he kept reliving the learning state he had just been in.
"That feeling of immersing yourself in learning is really addictive."
After finishing his meal, Lu Feng went straight back to the library.
The afternoon's lesson began with points.
Indefinite integrals, definite integrals, improper integrals, multivariable calculus... He spent more than half a year in his previous life barely understanding these concepts, but this afternoon he went through them all in one go.
Of course, this doesn't mean that you've mastered every single knowledge point, but at least you've built a framework and understood the core theorems. You can then reinforce your understanding by doing practice problems.
During the trip, Lu Feng also tried to stop and rest, but found that as long as he did not deliberately interrupt the journey, his body was not very tired.
This should also be a passive, side effect of "efficient learning".
The sunlight outside the window had turned orange at some point, shining obliquely on the pages of the book and giving the draft paper a warm hue.
Lu Feng put down his pen and looked up at the clock on the wall.
18: 02.
Lu Feng stretched his stiff neck and opened the system.
Current learning value: 1050
In one day, from 0 to 1050.
This efficiency completely exceeded expectations.
Without hesitation, Lu Feng opened the store and selected "Exchange for Black Technology Blueprints".
[Confirm redemption after consuming 1000 learning points?]
confirm.
The learning value jumped from 1050 to 50, and then a golden light lit up in the center of the panel.
A blueprint emerged from the light.
It hovered in front of Lu Feng, about the size of an A3 sheet of paper, its semi-transparent blue background densely covered with mathematical formulas and mechanical models.
The title boldly proclaims: "[Fractional Differential Proof of the Constitutive Equation for Elastic Materials]"
Lu Feng's breath hitched slightly.
He knew the constitutive equations.
This is the core equation describing the stress-strain relationship of materials, and one of the cornerstones of solid mechanics.
Classical Hooke's law, elastoplastic models, viscoelastic models... are all special forms of constitutive equations.
But fractional derivatives?
Traditional calculus deals with integer derivatives of the first, second, and third order.
Fractional derivatives extend the order of the derivative to any real number and even complex number domain.
The 0.5 derivative, the 1.7 derivative... these things that have no intuitive physical meaning in classical mathematics are rigorously defined within the framework of fractional calculus.
The mechanical behavior of many novel polymer materials and composites cannot be accurately described by classical integer-order equations; only fractional-order models can accurately approximate experimental data.
Lu Feng's gaze swept downwards from the headline.
"The first part is the mathematical foundation—the definition of the Riemann-Liouville fractional derivative, the definition of the Caputo fractional derivative, and the differences between the two in the treatment of initial conditions."
"The second part is physical modeling—replacing classical elastic and damping elements with fractional differential equations to construct a generalized Scott-Blair model."
"The third part is the core derivation—using the Mittag-Leffler function as a basis, the fractional constitutive equation is solved by Laplace transform, and finally a stress relaxation function containing the fractional parameter α is obtained."
"The fourth part is the conclusions and applications—when α=1, the equation degenerates into the classical Maxwell model; when α=0, it degenerates into the purely elastic Hooke's law."
Lu Feng felt increasingly uneasy as he looked at the document.
"This cutting-edge technology is incredibly advanced and powerful."
Even in the academic world of 2026, the content of this drawing would be considered a cutting-edge research direction.