No Room Left

"Keep it simple, Zhou Yu'an."

Professor Lu's patience was running out. "Keep it simple."

"Alright, alright, three sentences, the last three sentences!" After being scolded by Professor Lu, Zhou Yu'an finally got to the point. "Using the concept of important limits and the property of bounded variables multiplied by infinitesimals, combined with the squeeze theorem, we can find that this double limit is 0. That's my core idea, that's all."

"Alright, Zhou Yu'an, you can leave now," Professor Lu said sternly, then added, "Your algorithm and result are correct, but I can only give you 60 points. I deducted 40 points because you talked a lot of nonsense."

Zhou Yu'an left the stage dejectedly and returned to her seat, unhappy.

Professor Lu's lesson continues. The next question is a proof problem, given some simple conditions, requiring proof that there exist ζ, η∈(a, b) such that f'(ζ)=a+b/2ηf'(η).

Shao Tiantian went on stage to complete the verification; his department relied on him alone to support the entire department.

"... So I used two mean value theorems, Lagrange's mean value theorem and Cauchy's mean value theorem, to prove it." Shao Tiantian explained his proof in half a minute.

"Very good, concise and to the point." Professor Lu was very satisfied, Shao Tiantian's status in his mind continued to rise.

"I gave a few questions, and students such as Shen Qi, Shao Tiantian, and Zhou Yu'an all shared their opinions and provided some ideas. Here I will give a summary, and you can take notes." Professor Lu's teaching steps are to first have students do the questions and evaluate each other, and then he highlights the key points and gives a summary.

Professor Lu said, "Compared to other branches of mathematics, mathematical analysis is very young. Before the 19th century, it couldn't even be considered a branch. The first mathematicians to realize the need to inject rigor into analysis were Gauss and Abel, who even argued about it. After a heated debate, Abel fell seriously ill, died of depression, and was only 27 years old."

"The young mathematical genius Abel died young, and the great Gauss felt guilty. After all, Abel, a young genius of his time, died young, and the master Gauss bears some responsibility."

"Gauss lived to be almost 80 years old, still vigorous and in good health. In his later years, he wrote a monograph, 'Calculation of Calculus,' which we can consider the prototype of mathematical calculus. This was in the mid-19th century. So, as I said before, the collision of ideas generates the driving force for academic development." Professor Lu paused here.

The students in the audience listened with great interest. Gauss was indeed amazing; he had used his academic theories to make Abel so angry that he died. This was the powerful fighting force that only a master could possess.

The veracity of Professor Lu's anecdotal accounts of mathematics may need further verification, but students love listening to History of Mathematics; it's much more interesting than the dry theories in textbooks.

To liven up the class, Professor Lu started with a bit of mathematical anecdotes, then smoothly transitioned into the main topic. "Standing on the shoulders of giants, and with the further refinement by Cauchy and Weierstrass, Lebesgue completed the final work in the early 20th century, making * Mathematical Analysis * a globally recognized mathematics course, included in the foundational textbooks of the Mathematics Department in universities worldwide. In the following lessons, I will discuss Lebesgue integrals. Lebesgue, this Frenchman, also has many interesting stories worth mentioning."

"From the solutions and discussions of the previous questions, we discovered that between two limits, an infinitesimal increment of a variable always produces an infinitesimal increment of the function itself. In other words, f(x) is a continuous function of x in the neighborhood of a definite value of the variable x. A fundamental property of continuous functions is that they are insufficient to guarantee the continuity of the function."

"Dear students, please remember this fundamental property. It arose from the intellectual exchange among young mathematicians such as Shen Qi, Shao Tiantian, and Zhou Yu'an... I hope you will all become true mathematicians in the future," Professor Lu said with a smile.

Shen Qi, Shao Tiantian, and Zhou Yu'an also smiled, greatly encouraged, and the relationship between teachers and students became more harmonious amidst their laughter and conversation.

Other students gradually accepted and adapted to Professor Lu's teaching style. Only when you like a professor's class will you be interested in learning the course well, even if you don't understand it very well now, because interest is the best teacher.

"Alright, we still have some time, let's do a few more problems," Professor Lu said, writing new problems on the blackboard.

At the beginning of the class, some students were resistant to Professor Lu's style of giving questions out of the blue.

Now, everyone is eagerly awaiting the new questions, itching to try their best.

Professor Lu's influence was subtle and pervasive; in less than a class period, he transformed the students' initial resistance to his teaching into acceptance.

The new problem is to calculate I = ∫e^xsinydy - e^xcosydy.

"It's the Mathematics Department's turn again." Professor Lu looked at Shen Qi and understood. Shen Qi was the core figure and the Boss of the Mathematics Department. It seemed that Shen Qi had several capable subordinates. The Boss usually didn't take action easily. If there was a problem, he would send his subordinates to solve it first. Only when the subordinates couldn't handle it would the Boss step in.

Shen Qi turned to look at Zhou Yu'an and Ou Ye's positions, giving Ou Ye a meaningful look. "Calculation Ji, it's your turn this time."

Professor Lu followed Shen Qi's gaze and scanned the back seats, locking onto Ou Ye: "The previous few students were all male, now we'd like to invite a female student to come up on stage. Ou Ye, please come up."

Without wasting any words, Ou Ye stood up, went up to the stage, and answered the question on the blackboard with chalk.

Ou Ye quickly calculated the result: I = 1 - e^2.

"OK, Ou Ye, what was your thought process when you calculated this result?" Professor Lu asked.

Ou Ye replied, "Green's formula."

Professor Lu pressed further. "Be more specific, I need details, more details."

Ou Ye looked helplessly at Shen Qi, remaining silent.

Shen Qi knew it wasn't that Ou Ye didn't understand, but rather that she wasn't good at expressing herself.

Shen Qi stepped in to defuse the situation. "D is a closed curve bounded by L and L1. We can calculate a value e squared minus 1, and then use Green's theorem to finally get I equal to 1 minus e squared. This is my understanding of Ou Ye's approach."

Professor Lu asked Ou Ye, "Is that what you think too?"

Ou Ye nodded.

Professor Lu: "Then why didn't you say so yourself?"

Ou Ye: "I can calculate, but I can't explain."

Some students in the audience laughed. This girl is interesting; her calculations are sharp, but she doesn't speak fluently.

"Ou Ye, you can go back to your seat now. Your calculations are correct, but your language expression skills need further improvement," Professor Lu said.

"Alright, last question."

Professor Lu erased the blackboard, drew a curve, and posed the question: Please prove that: m/m+2∫dx/√[1+(x/a)^m]=arcPP1-(P1R1-PR)

The moment the question was posed, the audience fell into complete silence.

"The last question is for the Department of Scientific and Engineering Computing," Professor Lu said, looking at Shao Tiantian.

This time, Shao Tiantian did not go on stage immediately. He was confused and had no idea how to prove it.

Not a single person from the Department of Scientific and Engineering Computing stepped forward. It's easy to act tough, but to truly show off, you need top-tier skills. Without those skills, you can only watch helplessly.

"What about the Mathematics Department?" Professor Lu asked Shen Qi.

Shen Qi stood up. This time, he wouldn't send his underlings. He knew that he was probably the only one in the entire Mathematics Department who could provide a complete proof for this problem. If there was a second person, it would be Ou Ye, but the derivation and proof of this problem would be very complicated. Given Ou Ye's style of expression, she wouldn't be able to explain the proof's logic even if she talked for three days and three nights.

"Shen Qi, you're here?" Professor Lu asked.

"I'll do it." Shen Qi went up to the stage, picked up a new piece of chalk, and began to derive and prove the derivation on the blackboard.

"PR and P1R1 are the tangents to the curves at points P and P1, respectively. So, I will calculate the difference between the two definite integrals..." Shen Qi wrote and spoke at the same time.

Therefore: arcQQ1-arcPP1=(Q1S1-QS)-(P1R1-PR)

"In dealing with ellipses, I use algebraic expressions to represent the difference of infinitely many arc segments. The analysis is as follows..."

∫Xdx+∫Zdz=-hxz/√[-fl]

"The proof of this problem is quite complicated, let me think about it." Shen Qi wrote on half of the blackboard and paused briefly.

The students in the audience, including Shao Tiantian and Zhou Yu'an, who were praised by Professor Lu as "young mathematicians," were dumbfounded. They couldn't quite understand Shen Qi's reasoning and proof.

Professor Lu remained calm and observed the situation.

"I've got it! Here I'll use a geometric interpretation to make this formula consistent with the integral, where p is the positive focal chord of the ellipse..."

After a moment's thought, Shen Qi continued to verify: arcJD + arcDG =...

His idea was to let x=0, then arc JD would disappear, and the algebraic terms in equation (7) would also disappear, so arc DG would become arc DA... Shen Qi quickly filled the blackboard with his writing.

"It's a very old proof method, the Fagnano's Theorem, it's very classic." Professor Lu was a little surprised that Shen Qi could grasp the core idea of Shen Qi's derivation; Shen Qi actually used this approach to prove it.

"So, I'll add... wait, there's no room left." As Shen Qi wrote, he realized that the entire blackboard was filled with his writing, leaving no room at all.

Shen Qi turned around and threw half a piece of chalk into the blackboard slot. "I'm pretty sure this equation is true, but there's not enough blank space on the blackboard to write it down."

The audience was initially stunned, then realized that two or three hundred years ago, a French amateur mathematician named Fermat had done the same thing.

"I was pretty sure the assumption was true, but there wasn't enough space in the book to write it down." That's how Fermat's Last Theorem came about, and it wasn't until 1995 that Wiles proved it.