28: Chapter 28 Advanced Mathematics, First Stage Closed Loop

Systems, when written on paper, always seem exceptionally reliable.

Jiang Lin even had a misconception at the very beginning.

It seemed as if as long as he wrote down the schedule, the undergraduate training plan for the School of Physics at Peking University that he had scraped from the internet was already a small fraction complete.

Of course, he also knew in his heart that it was impossible for him to truly replicate a Peking University classroom.

There were no teachers, no recitation classes, no peer discussions, and no laboratories.

All he could get were public training programs, dozens of downloadable electronic versions of textbooks, unsystematic public course videos, and a bunch of online course resources of unknown quality.

Following the path of the Peking University Physics department was essentially just finding himself a signpost that was tall enough, hard enough, and incapable of deceiving himself.

People are always like this.

When making a plan, it feels as if one has already completed it.

Writing getting up at six on paper is as if he truly won't stay in bed at six tomorrow morning.

Writing Advanced Mathematics at eight in the morning is as if that four-hundred-and-twenty-eight-page textbook has automatically become a little thinner.

Writing review and Time Capsule at seven in the evening is as if his future self at thirty, forty, and sixty years old is already standing in the distance nodding to him.

But at six the next morning, the Wasteland dragged Jiang Lin out of this illusion in an extremely rustic way.

He didn't hear the alarm clock.

It wasn't that the alarm clock was broken.

It was that he was woken up by the cold before the alarm clock rang.

No matter how the Stone House was plastered, it always leaked wind.

The night wind drilled in through crevices from God knows where, like tiny files scraping against his head and face.

When Jiang Lin opened his eyes, the room was as dark as a well.

Lying in his sleeping bag, his first reaction wasn't to get up, but to pull his head back in.

It was too cold, and his body simply didn't want to move.

The moment his fingers stretched out of the sleeping bag was like thrusting them into a basin of ice water.

If it were in the Real World, he could probably laze around for five minutes today.

Five minutes after five minutes.

Anyway, there would be Mom making breakfast in the kitchen, Old Liu patrolling the classes at school, and the morning reading bell urging for his life over and over in the hallway.

But the Wasteland didn't have these.

The Wasteland would only quietly watch him rot away.

Jiang Lin kept his eyes open in the dark, staring at the Stone House ceiling where nothing could be seen clearly, lying there for a full three minutes.

Then he cursed in a very low oath, gritted his teeth, and crawled out of the sleeping bag.

This morning, he did not start farmland maintenance at exactly six o'clock as written on the schedule.

He only walked to the edge of the field at six-twenty-seven.

The wind blew from the direction of the dried-up riverbed, carrying the mineral scent left behind after the acid rain.

A layer of mud and dust covered the potato leaves, the Soybean seedlings were also somewhat withered, and the Pumpkin vines shrank by the edge of the ridge like a few frozen snakes.

Jiang Lin squatted down and touched the surface layer of the soil with his fingers.

Hard, icy cold.

Like a piece of old iron that hadn't completely woken up yet.

He picked up a small shovel, clearing out the soil blown back to the edge of the drainage ditch by the wind shovel by shovel, and checked several windbreaks pinned down by stone flakes.

When he returned to the Stone House, a large patch of dark red light appeared on the horizon.

The time was already seven-fifty-two.

There were only eight minutes left until the eight o'clock Advanced Mathematics in the morning.

Jiang Lin stood at the doorway, looked at his hands covered in mud and dust, and then looked at the schedule on the wall.

At this moment, he understood for the first time that systems were not written for times when blood was boiling.

Systems were written for times when one didn't want to move, didn't want to study, and didn't want to live with such forceful effort.

Therefore, he didn't change the schedule or find excuses for himself.

He just washed his muddy hands clean, drank a sip of warm water, sat down in front of the stone table, and opened the textbook.

Of course, the Physics department of a truly top-tier university would not understand mathematical foundations so simply.

Mathematical Analysis, Linear Algebra, Analytic Geometry, Ordinary Differential Equations—none of these were thresholds that could be easily crossed.

Yet he currently found it difficult even to stand firmly.

For someone from an ordinary high school without teachers, who could only rely on electronic textbooks and public videos for self-study, Advanced Mathematics was not the destination, but the first handrail he could touch.

On the first morning, he only read two pages.

One of the pages was still the definition of function limits he had already read in the previous chapter.

He took out his concept notebook and opened a new column behind the page on limits.

["Sequence Limit: Go forward, find N."]

["Function Limit: Shrink inward, find δ."]

["Commonality: Others give the error, I provide the control."]

["Difference: One controls the sequence number, the other controls the distance."]

After writing these few lines, he added another sentence.

["Don't think you understand them just because they look similar."]

On the twenty-third day, Jiang Lin was pressed back to the surface of the earth by derivatives.

During high school, he had studied derivatives.

He had memorized differentiation formulas, worked on monotonicity, and written tangent equations.

In Gaokao questions, derivatives were a weapon, the sharpest and most intimidating blade in the final pressing question.

But the derivatives in college textbooks were not quite the same as that blade in his memory.

It didn't immediately ask you to find the maximum value of a certain function, nor did it ask you to prove a certain inequality.

It first asked you a colder question.

What is called the rate of change at a certain point?

What is called local linear approximation?

What is called a curve being approximately replaced by a straight line in a sufficiently small neighborhood?

Local, linear, approximation.

He recognized every single word.

Put together, he didn't recognize them anymore.

That morning, Jiang Lin sat in front of the stone table for four hours and only did one thing.

Drawing tangents.

He drew curves on paper.

Drawn quite poorly.

Some looked like curved vines, some like sand ridges gnawed out by the wind on the Wasteland, and some like irregular cracks on the edge of a puddle.

Then he selected points on the curve.

Drawing tangents bit by bit.

For the same function at different positions, the tangent directions were different.

Near the same point, magnifying the curve, magnifying it again, and magnifying it yet again, it seemed to truly look more and more like a straight line.

By the end of the drawing, the handwriting tablet was covered everywhere with crooked lines, like a dead forest blown down by a storm.

He knew how to calculate the answers.

At least following the example problems, he could calculate the answers.

But he didn't dare say he understood.

When patrolling the fields in the afternoon, Jiang Lin didn't immediately return to the house to study General Physics.

He stood by the ridge of the field, watching the numbers on the thermometer rise bit by bit.

At six in the morning, eleven degrees.

At eight in the morning, fourteen degrees.

At ten, twenty degrees.

At noon, twenty-six degrees.

The sun climbed higher little by little, and the surface temperature of the Wasteland also changed bit by bit.

In the past, he recorded these numbers just to judge when to water, when to cover with windbreak film, and when to avoid the strongest sunlight.

That day, staring at that string of temperatures, he suddenly realized something.

Temperature itself was not a derivative.

The speed at which temperature changed was.

Rising from eleven degrees to fourteen degrees, if it took two hours, that was one kind of change.

If it took twenty minutes, that was another kind of change.

Walking from the Stone House to the puddle, walking slowly, the soil underfoot became wet bit by bit.

Walking fast, the dampness and sour smell almost slammed into his face.

Location wasn't the key point; how the location changed was the key point.

Jiang Lin stood by the edge of the field, the wind making his lips dry.

He suddenly turned around and ran back to the Stone House, writing down on the draft paper:

["Derivative is not position"]

["Derivative is the speed of change"]

After writing, he looked at it for two seconds and felt it wasn't enough.

So he added a line below.

["Derivative is not where I am right now, but how I am leaving this place."]

He realized later that the so-called velocity and acceleration in General Physics had actually been standing right here waiting for him all along.

The rate of change of position with respect to time was velocity, and the rate of change of velocity with respect to time was acceleration.

He had memorized these words in high school, but until this day, he felt for the first time that those formulas were not conclusions printed in books, but bones grown out of change.

In that night's Time Capsule, he wrote down his first truly ugly record.

["Knowing how to differentiate does not equal understanding derivatives; don't deceive yourself."]

On the fifty-second day, he began to study integration.

This time, Jiang Lin did not repeat the same mistake.

He didn't start by memorizing formulas.

Nor was he in a hurry to look at those example problems with long radical signs and trigonometric functions.

He first walked to the edge of the farmland and divided a piece of land into many narrow strips.

Each strip was not completely regular.

Some were a bit wider, some a bit narrower, some had their edges bitten off by wind erosion, and some places couldn't even be considered effective arable land because there were too many stones.

If he estimated it using a single rectangle, the error would be too large.

If cut into ten pieces, it would be a bit more accurate.

Cut into one hundred pieces, it would be even more accurate.

What if he cut it until the end, when every small piece was so small that it could almost be regarded as a regular shape, and then added them all together?

He even used a tape measure to measure the width of several ridges of land, crudely cut the irregular edges section by section, and then estimated them using approximate trapezoids.

The numbers calculated were, of course, inaccurate.

The ridge was not a standard curve, and the wind-eroded gaps wouldn't obediently grow into the figures in the textbook.

But at that moment he suddenly understood that integration was never useful because the world was regular; precisely because the world was irregular, humanity needed this clumsy method.

Jiang Lin squatted by the edge of the field, holding a withered branch, and drew on the ground for a whole morning.

By the end of his drawing, looking at those dense, tiny strips, he suddenly laughed.

Integration was not a mysterious symbol.

Integration meant admitting that the world was irregular, and then using a sufficient number of sufficiently small regular fragments to piece it back together.

After that day, another sentence appeared on the wall of the Stone House.

["Differentiation is taking apart."]

["Integration is putting together."]

Jiang Lin stared at this sentence for a moment, feeling it was too beautiful.

So beautiful that it felt like deceiving himself.

So he added another line below it.

["Disassembled finely enough, the assembly will be accurate enough."]

On the seventy-ninth day, he used the idea of integration for the first time to estimate the effective power generation of the solar panels in a day.

Early in the morning, the power output was very low.

In the morning, the power output rose.

At noon, it reached its peak.

In the afternoon, as the sun tilted, it dropped bit by bit.

In the past, he only looked at how much electricity the storage battery gained in the end.

Today it increased by seventeen percent, tomorrow by twenty-one percent, and the day after tomorrow was cloudy, increasing by only six percent.

These numbers were very important for survival.

But they were just results.

Now, he began to draw curves.

The horizontal axis was time.

The vertical axis was power.

That section in the early morning clung to the bottom, like a snake that had just woken up.

Slowly raising its head in the morning.

Swelling into an irregular peak at noon.

Collapsing bit by bit again in the afternoon.

The irregular area beneath the curve was the electricity that truly fell into the storage battery that day.

He didn't have a real power recorder, so he could only check the instantaneous power on the controller every half hour, and then connect these points into a rough broken-line graph.

Once at six-thirty in the morning, once at seven, and once at seven-thirty.

Between every two adjacent time points, he estimated it using a trapezoid.

The base was half an hour, and the height was the average of the two power readings before and after.

Adding up more than twenty trapezoids a day, what he got was a power generation estimate that couldn't be any rougher.

Later, when he compared this number with the change in the storage battery's power, he found that on the days with the largest errors, it was often not cloudy days, but rather when the midday sun was too fierce, causing the solar panels to overheat and their actual efficiency to drop severely.

After finishing drawing that chart, Jiang Lin stared at it for a long time.

Then he wrote beside it:

["Electricity does not come all at once."]

["Knowledge does not come all at once either."]

["They are all accumulated bit by bit."]

This sentence was a bit pretentious.

He originally wanted to cross it out.

But the pen tip hovered over the paper, paused for a moment, and ultimately put it down.

After staying in the Wasteland for a long time, one should allow oneself to be occasionally pretentious.

On the one hundred and thirty-sixth day, Jiang Lin was mentally broken by series once.

The wind was very strong that day.

The dust pouring through the gaps in the Stone House door dyed the edges of the draft paper dark red.

Jiang Lin stared at the infinite series in the textbook for a full two hours, and only one thought remained in his mind.

Why, having already had limits, derivatives, and integrals, was there still a need for infinite summation?

The two words infinity itself made people irritable.

It wasn't like a field; once tilled, it was tilled.

Nor was it like a meal; once eaten, it was eaten.

It was like the horizon of the Wasteland.

You walk over, and it's still there.

You walk further, and it's still there.

You think you have approached the finish line, but the finish line has merely changed positions to keep watching you.

That day, he didn't finish his plan.

During the four hours in the morning, he only understood two pages.

In the afternoon, he couldn't get into studying Physics either.

The first chapter of mechanics was about particle motion, but looking at those velocity, acceleration, and displacement formulas, he only felt they looked like a bunch of bugs crawling on paper.

During the evening review, he stared at the textbook for a long time.

According to the original plan, he should write down today's gains.

[part:gemini-3.6-flash]

Partial sums, convergence and divergence, the comparison test, the ratio test, the root test, the alternating series test.

It was just that looking at each test individually, he felt like he understood it.

But as soon as the problem changed, he immediately didn't know where to begin.

Some series looked like they should use the ratio test, only for him to get stuck on the limit halfway through.

Some looked comparable, but he couldn't find a suitable comparison series.

Some clearly had terms getting smaller and smaller, yet in the end they could still diverge.

The most disgusting part was that he couldn't just rely on his high school problem-type intuition to force his way through anymore.

Series didn't work that way.

So he couldn't write down anything useful.

In the end, he only wrote a single line.

[Lost today.]

After writing those words, he set his pen down, sat at the doorway of the Stone House, and watched the sky darken bit by bit.

No one in the Wasteland comforted him.

No one would say, "You've already worked very hard."

No one would say, "Take your time."

Nor would anyone say, "You're doing great."

The Wasteland would only blow wind into his collar, letting the night cold seep into his bones bit by bit.

In the distance, the dried-up riverbed was left as nothing more than a deeper shadow in the darkness.

The potato leaves in the field were pressed low by the wind, then slowly rose back up.

Jiang Lin sat for a long time.

Long enough for his fingers to grow numb from the cold before he finally returned to the Stone House.

At six o'clock the next morning, he got up as usual.

Checked on the potato sprouts as usual.

Inspected the filtration device by the puddle as usual.

Straightened out the Pumpkin vines blown askew by last night's wind as usual.

Sat back down at the stone table as usual.

Beneath yesterday's line "[Lost today.]", he wrote another line:

"[But the match isn't over.]"

Then he reopened the chapter on series.

Later on, series didn't become easier.

Jiang Lin was just gradually becoming less afraid of them.

He began to understand that so-called infinite summation did not equal chaos.

Some infinities would explode, while others would converge.

Some things seemed like they could never finish adding up, yet in the end, they would be contained within a finite bound.

This was very much like himself.

During his first time in the Wasteland, he barely survived a day and a night in total disarray.

The second time in the Wasteland, he scraped by on compressed biscuits and water until his supplies ran out.

The third time in the Wasteland, he took nine years to turn two hundred square meters of red soil into a field barely capable of sustaining life.

The fourth time in the Wasteland, he sat in the Stone House, doggedly wrestling with a copy of Advanced Mathematics.

His life also looked like a messy sequence of terms.

Chaotic, full of failures and fluctuations early on, even entirely lacking pattern.

But as long as it could keep moving forward, it wasn't considered divergent.

In the spring of the second year, Jiang Lin opened Advanced Mathematics (Part 2).

Multivariable functions were even better suited to the Wasteland than he had imagined.

Because the Wasteland never gave him single-variable problems.

A reduced potato yield wasn't necessarily due to a lack of water.

It could be acid rain, sunlight, or temperature.

It could be that too much plant ash had been scattered, or that the seeds had degenerated.

Or perhaps the concentration of a certain mineral in the groundwater had suddenly spiked.

In the high school problems of the Real World, questions were always clean.

Giving you conditions, asking for your answer.

But the Wasteland wasn't like that.

The Wasteland threw all the variables into his face all at once.

Moisture was changing, temperature was changing, soil was changing, seeds were changing, and his own physical stamina and judgment were changing as well.

Sometimes, even when he followed the exact method that succeeded the previous year, the harvest would still fall short by a noticeable margin.

Sometimes, if he merely drained water two days earlier, a few specific rows of potatoes would actually grow better.

Sometimes, after a spell of acid rain, the edges of the Pumpkin leaves would turn scorched and yellow, making him think it was over, only for new leaves to sprout again half a month later.

The Real World was never a single-variable problem.

Thus, for the first time, Jiang Lin truly grew fond of partial derivatives.

Not because it was easy to calculate, but because it was honest.

It acknowledged that the world was complex, and that a person could only clearly see one direction at a time.

Holding other variables constant, seeing only how changes in moisture affected yield.

Holding moisture constant, seeing only changes in soil pH.

Holding pH constant, seeing only sunlight.

Of course, the Wasteland wasn't a laboratory.

He couldn't truly hold all other variables constant.

There were no standard soil samples, no control groups, no precision sensors, and he didn't even know the mineral composition of each acid rain spell.

All he could do was a clumsy form of field approximation—changing only the watering frequency this year while keeping other practices as fixed as possible, then changing only the amount of plant ash next year, while logging yield, leaf condition, spot count, and soil texture item by item.

This was not a rigorous experiment.

But for him, this was already scraping order out of chaos.

Besides, approximation is the first foothold humanity manages to secure for itself when facing a complex world.

That evening, he wrote in his concept notebook:

"[Multivariable functions: Acknowledging that the world has more than one cause.]"

"[Partial derivatives: Temporarily pretending other things stay unchanged to clearly see one direction first.]"

"[This isn't the entirety of reality, but it is the first step toward understanding reality.]"

In the winter of the second year, he learned about multiple integrals for the first time.

The snow fell very lightly that day.

Strictly speaking, maybe it couldn't even be called snow.

The stuff falling over the Wasteland was somewhere between ice pellets and dust, hitting the roof of the Stone House with a fine, rustling sound.

Sitting inside, Jiang Lin looked at the double integrals and triple integrals in the textbook, and his farm field suddenly floated into his mind.

Single integrals were like walking along a line, double integrals like spreading across a patch of land, and triple integrals like slicing space itself into block after block.

He suddenly felt that mathematics possessed an ambition that was almost cruel.

It wasn't satisfied with calculating a single path; it wanted to calculate a field.

And it wanted to calculate a mass of air, a rock, a puddle, an entire world filled with variables.

In the spring of the third year, Jiang Lin studied line integrals and surface integrals.

He studied those chapters extremely slowly.

So slowly that he once wondered if he had returned to the days when he first encountered ε-δ.

Path, direction, flux, circulation.

Every single word felt like it was covered in thorns.

He tried to understand them through the lens of the Wasteland.

The wind passing through the doorway of the Stone House was a flux.

Water flowing down a drainage ditch was accumulation along a path.

Sunlight falling on a tilted solar panel wasn't as simple as just hitting it; it had to do with angles as well.

Some things weren't about how strong they were at a single point, but about what they accumulated after passing through an entire surface or along an entire path.

He studied with great pain.

But this pain was different from that of the first year.

The first year's pain was panic, incomprehension, and doubting whether he was even qualified to open the book.

The third year's pain was more like carrying a heavy load.

He knew he walked slowly, but he also knew he was indeed moving forward.

In the winter of the third year, the walls of the Stone House were almost completely covered with paper.

His concept notebooks spanned seven volumes.

The handwriting in the first volume was the largest and heaviest, like someone who had just picked up a weapon and wanted every blow to make a loud noise.

Starting from the third volume, the handwriting stabilized.

After the fifth volume, there was noticeably less nonsense.

By the seventh volume, he rarely wrote "I finally understand."

He only wrote:

"[Still needs review here.]"

"[Re-check in three months.]"

"[Can be applied to the Electromagnetism section of General Physics.]"

"[Related to line integrals and flux concepts, leave a placeholder for now.]"

"[Don't rush to declare victory.]"

He no longer easily declared that he understood.

Perhaps this in itself was a kind of progress.

During these years, it wasn't that he hadn't touched General Physics at all.

It was just that every time he opened it, he was forced back by his own mathematical shortcomings.

Seeing velocity and acceleration, he knew he had to go back to review derivatives; seeing work and energy, he knew he had to go back to review integrals; seeing electric fields, magnetic fields, and flux, he knew that sooner or later he would have to face line and surface integrals.

Time and time again, he merely realized that his past ability to "do problems" was often nothing more than putting formulas in the right places.

In fact, Advanced Mathematics hadn't been conquered by him either.

Jiang Lin later grew to hate the word "conquer."

How could a person conquer mathematics?

After being repeatedly beaten by it, he had simply learned to stand firm, to guard his vital spots, and to no longer just close his eyes when the next punch came.

In the summer of the fourth year, he began his second pass.

The first pass was reading the textbook; the second pass was solving problems.

This time, the pain took a different form.

While reading, he could comfort himself that going slow was fine as long as he was understanding.

While doing problems, wrong was wrong, and being unable to solve it was being unable to solve it.

If a proof couldn't be written out, it just couldn't be written out, leaving no leeway whatsoever.

"[Mistook a necessary condition for a sufficient condition.]"

"[Forgot to check the domain.]"

"[Conditions for swapping limits were not satisfied.]"

"[Integration bounds were not changed after substitution.]"

"[Mechanically copied example problems here without understanding.]"

...

In the autumn of the fifth year, the first cold wind swept across the dried-up riverbed and blew into the Stone House.

That morning, he got up at six as usual.

In the field, the potato leaves were pressed low by the wind, then slowly sprang back up.

Pumpkin vines lay flat along the edge of the field, with a few withered Pumpkins hidden beneath the leaves.

The Soybean pods were half dried, making a faint rustling sound at the gentlest touch.

Having checked the field, Jiang Lin washed his hands and returned to the Stone House.

Advanced Mathematics (Part 1) and Advanced Mathematics (Part 2).

He had flipped through the electronic editions of both volumes an untold number of times.

His notes were stacked under the stone table, tied with hemp rope into several piles like bricks.

He sat down and opened to the last few review problems in the final chapter.

There were no dramatic difficulties, nor was there any sudden ultimate problem that would stump him for three months.

Just a few comprehensive problems.

Limits, partial derivatives, multiple integrals, series, line integrals.

Knowledge points intertwined together, like streams merging in a low basin.

Jiang Lin wrote very slowly.

Checking every step, reviewing every condition.

Starting at eight in the morning, at 10:46 AM, he wrote the final line.

Proof complete.

Outside, the wind was still blowing; inside, it was quiet.

There were no cheers.

Nor was he pacing excitedly back and forth in the Stone House as he had once fantasized.

In that instant, he remembered his bewilderment when he first saw ε.

He remembered the meager yet solemn sense of achievement when he first proved that 1/n approaches 0.

He remembered the afternoons spent being ground into the dust by derivatives, the page where he cursed at integrals, and the "[Lost today.]" he wrote on the day of series.

And he remembered the next morning's "[But the match isn't over.]"

As it turned out, so-called "finishing" wasn't suddenly becoming omnipotent at a single moment.

It wasn't that after closing the book, the entire content flowed through one's brain like clear water.

It wasn't seeing through every problem at a glance and having every concept at one's fingertips from then on.

Finishing simply meant that he had finally walked across this Wasteland himself.

Where the pits were, where the slopes were, where one would get lost, and where one needed to look back—he had left footprints across it all.

In some places, he was still unfamiliar; some theorems he might still forget after a few years.

Some proofs he could write out now, but if presented in another form, he would still get stuck.

But this time, he was no longer standing outside the door.

He had finally come inside.

Jiang Lin opened the Time Capsule and wrote on the last page:

"[Advanced Mathematics, Phase 1 loop closed.]"

After finishing, he paused for a long time and added another line:

"[Not total mastery, but the textbook's main storyline and the first round of problems have finally closed the loop.]"

As his pen lifted, Jiang Lin began to laugh.

Starting today: Mechanics, Electromagnetism, Thermodynamics...

Those things that once weighed upon the horizon like distant black mountains were finally no longer just dark shadows.

He still couldn't see all the mountain paths clearly, nor had he even finished measuring the true height of the very first mountain.

But he already possessed a pair of feet capable of walking mountain paths.

Content that college students in the Real World might finish in a single year had been forcibly dragged out into five years by him.

It sounded very slow—slow to the point of near foolishness.

But college students in the Real World didn't need to fix drainage ditches every day, didn't need to maintain solar panels, didn't need to protect two hundred square meters of farmland amidst acid rain and cold winds, and didn't need to face a silent Wasteland all alone.

More importantly, he was not a genius.

He was merely a person who forcefully crammed not knowing, forgetting, misunderstanding, and starting over into time.

At 1:00 PM, Jiang Lin decided to turn back to the first page of General Physics.

Not to start from scratch, but to go back for a postmortem.

He wanted to see how many places over the past five years he merely knew how to solve, rather than truly understood.