I Just Want to Be a Quiet Top Student

"Easy, Zhou Yu'an."

Professor Lu's patience was about to run out: "Speak in a simpler way."

"Okay, okay, three sentences, the last three sentences!" Zhou Yu'an was scolded by Professor Lu, and finally said the key point: "Using the idea of ​​​​important limits, and the nature of bounded variables multiplying infinitesimal quantities, combined with the two-sided clamping theorem, find So this double limit is 0. This is my core idea, it’s over.”

"Okay, Zhou Yu'an, you can go down." Professor Lu said with a straight face, and then added: "Your algorithm and results are correct, but I can only give you 60 points, and you will be deducted 40 points because of your nonsense. .”

Zhou Yu'an resigned from the stage and returned to his seat, unhappy.

Professor Lu's teaching is continuing. The next question is a proof question. Some simple conditions are given to prove the existence of ζ, η∈(a, b), so that f'(ζ)=a+b/2ηf'(η )

Shao Tiantian came to the stage to complete the proof, and they relied on him alone to support the overall situation.

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

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

"I came up with a few questions, Shen Qi, Shao Tiantian, Zhou Yu'an and other students all expressed their opinions and provided some ideas. Here I will make a summary, and students can remember it." Professor Lu's teaching steps are, First let the students do the questions and evaluate each other, and then he draws the key points and makes a summary.

Professor Lu said: "Compared with other branches of mathematics, fractions are very young. Before the 19th century, it could not even be regarded as a branch. The first mathematicians who realized the need to inject rigor in analysis were Gauss and Abel. There were fights about it. After a heated debate, Abel suffered a serious illness and died of depression at the age of 27."

"The young mathematical genius Abel died young, and the great Gauss felt guilty. After all, he was mad at the young genius Abel of the same generation. The great master Gauss bears some responsibility."

"Gauss lived until he was almost 80 years old. He was strong and in good health. He wrote a monograph "Calculus Computation" in his later years. We can think that this is the embryonic form of fractions. This is the middle of the 19th century. So it is still the same sentence , The collision of ideas generates the impetus for academic development." Speaking of this, Professor Lu paused.

All the students in the audience listened with gusto. Sure enough, Gauss was the best. He made Abel mad with his academic theories. This is the powerful fighting power that only a master possesses.

Perhaps the authenticity of Professor Lu's wild history of mathematics needs to be further confirmed, but students love to listen to the history of mathematics, which is much more interesting than the boring theories in textbooks.

Lectured on the wild history of mathematics to mobilize the classroom atmosphere. Professor Lu entered the topic freely: "Standing on the shoulders of giants, after further improvement by Cauchy and Weierstrass, in the early 20th century, Le Besgue After completing the final work, "Mathematical Analysis" has become a worldwide mathematics course, which has been compiled into the basic textbooks of mathematics departments in various universities around the world. In the next few lessons, I will talk about Lebesgue integral, Lebesgue is a French People also have many interesting stories, which are worth mentioning."

"From the answers and discussions of the few questions just now, we found that between two limits, an infinitesimal increment of a variable always produces an infinitesimal increment of the function itself, in other words, f(x) in an infinitesimal increment of the variable x A fundamental property of continuous functions is not sufficient to ensure the continuity of the function to determine that the neighborhood of values ​​is a continuous function of x."

"Students, please remember this basic nature. It was born in the collision of ideas of young mathematicians such as Shen Qi, Shao Tiantian, Zhou Yu'an... I hope you can become real mathematicians in the future." Professor Lu laughed.

Shen Qi, Shao Tiantian, and Zhou Yu'an also laughed, and were encouraged. The relationship between teachers and students tended to be harmonious during the conversation and laughter.

Other students have also gradually accepted and adapted to Professor Lu's teaching methods. Only when they like a professor's class will they become interested in learning this course well. Even if they don't understand much now, interest is the best teacher.

"Okay, there is still some time, let's do a few more questions." Professor Lu said, writing new questions on the blackboard.

At the beginning of this class, some students were very repulsed by Professor Lu's style of asking questions whenever they disagreed.

But now, everyone is eagerly waiting for new questions, eager to try.

Professor Lu moisturizes things silently, and in less than a class, the students go from rejecting him to accepting him.

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

"This time it's the Mathematics Department's turn again." Professor Lu looked at Shen Qi, and he finally understood that Shen Qi is the core figure and boss of the Mathematics Department. It seems that Shen Qi has several strong generals under his command, the boss usually doesn't take action easily, if there is a problem, he will send the younger brother to solve it first, and if the younger brother can't figure it out, it is the boss' turn to come forward.

Shen Qi turned his head to look at the positions of Zhou Yu'an and Ouye, and gave Ouye his eyes: Ji Ji, it's your turn this time.

Professor Lu followed Shen Qi's gaze to scan the seats in the back row, and locked on to Ouye: "The first few are boys solving problems. Next, we invite a girl to come on stage. Ouye, please come on stage."

Ouye didn't talk nonsense, got up on stage, and took chalk to answer on the blackboard.

Soon, Euler calculated the result, I=1-e^2.

"OK, Ouye, on what basis did you calculate this result?" Professor Lu asked.

Ouye replied: "Green's formula."

Professor Lu asked: "Be specific, I need details, more details."

Ouye looked at Shen Qi helplessly, without speaking.

Shen Qi knew that it wasn't that Ouye didn't understand, but that she was not good at expressing it.

Shen Qi stepped forward to make a rescue: "D is a closed curve surrounded by L and L1. It is possible to calculate the square of a value e minus 1, and then use Green's formula to finally get I equal to the square of 1 minus e. This is my opinion. The understanding of Ouye's thinking."

Professor Lu asked Ouye: "You think so too?"

Ouye nodded.

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

Ouye: "I can count, but I can't tell."

Some students in the audience laughed. This girl is a bit interesting, she has sharp calculations, and she speaks clumsily.

"Ouye, go back to your seat first. Your calculation is correct, and your language skills need to be further strengthened." Professor Lu said.

"Okay, last question."

Professor Lu wiped the blackboard clean, drew a graph, and asked a question, please prove: m/m+2∫dx/√【1+(x/a)^m】=arcPP1-(P1R1-PR)

As soon as this question came out, there was a dead silence in the audience.

"The last question is for the Department of Science and Engineering Computing." Professor Lu looked at Shao Tiantian.

This time Shao Tiantian didn't come to the stage immediately, he was confused, he didn't have any ideas, he didn't know how to prove it.

No one from the Department of Science and Engineering Computing stepped forward. It was easy to pretend to be big, but it relied on top-level strength to pretend to be big.

"What about the Department of Mathematics?" Professor Lu looked at Shen Qi.

Shen Qi stood up, this time he didn't send his younger brothers and sisters out, he knew that he was probably the only one in the entire mathematics department who could give a complete proof of this question. If there is a second one, it is Ouye, but the derivation and proof of this question will be very cumbersome. With Ouye's language expression style, she can't finish the proof of ideas in three days and three nights.

"Shen Qi, are you coming?" Professor Lu asked.

"I'll do it." Shen Qi came to the stage, picked up a new piece of chalk, and deduced and proved it on the blackboard.

"PR and P1R1 are the tangents of the curve at points P and P1 respectively, so I will take the difference of the two definite integrals..." Shen Qi wrote while writing.

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

...

"For the processing on the ellipse, I use an algebraic expression to express the difference of infinitely many arcs, then, the analysis is as follows..."

∫Xdx+∫Zdz=-hxz/√【-fl】

...

"The proof of this question is quite troublesome, let me think about it." Shen Qi wrote half of the blackboard and paused for a while.

The audience, including Shao Tiantian, Zhou Yu'an and other outstanding students who were called "young mathematicians" by Professor Lu, were also dumbfounded. They couldn't quite understand Shen Qi's derivation and proof ideas.

Professor Lu remained on the sidelines quietly.

"I figured it out, here I quote the geometric meaning, make this formula consistent with the integral, p is the positive focal sine of the ellipse..."

After thinking for a while, Shen Qi continued to seek proof: arcJD+arcDG=...

His idea is to set x=0, then the arc JD disappears, and the algebraic term in the formula (7) also disappears, so the DG arc becomes DA arc...Shen Qi quickly filled the blackboard.

"A very old proof method, Fagnano's theorem, is very classic." Professor Lu was able to get the core idea of ​​Shen Qi's derivation. He was a little surprised that Shen Qi actually used this method to prove it.

"So, I will order again... Hey, there is no space." Shen Qi wrote and found that the entire blackboard was filled by him, and there was no room left.

Shen Qi turned around and threw half of the chalk into the slot on the blackboard: "I'm sure this equation is true, but there is too little space on the blackboard to write it down."

Everyone in the audience was confused at first, and then realized that two or three hundred years ago, a French amateur mathematician named Fermat did the same thing.

"I'm pretty sure the hypothesis is true, but there's too little room in the book to write about it." That's how Fermat's last theorem came about, and it wasn't proven until 1995 by Wiles.