"Keep it simple, Zhou Yu'an. ”

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

"Okay, good, three sentences, the last three sentences!" Zhou Yu'an was scolded by Professor Lu, and finally said the point: "Using the idea of important limits, as well as the property of multiplying bounded variables by infinitesimal quantities, combined with the two-sided clamping theorem, this double limit is 0." That's my core idea, that's it. ”

"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 I deduct you 40 points because you talk nonsense." ”

Zhou Yu'an stepped down and returned to his seat in a huff, unhappy.

Professor Lu's teaching continues, and the next question is a proof problem, which gives some simple conditions, which requires the proof of the existence of ζ, η∈(a,b), so that f'(ζ)=a+b/2ηf'(η)

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

“...... So I used two median theorems, the Lagrangian median theorem and the Cauchy median theorem, and I got it. Shao Tiantian spent half a minute explaining his proof ideas.

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

"I asked a few questions, and Shen Qi, Shao Tiantian, Zhou Yu'an and other students all expressed their opinions and provided some ideas. Here I will make a summary for students to memorize. "Professor Lu's teaching steps are to first ask students to do questions and evaluate each other, and then he draws the key points and makes a summary.

"Compared to other branches of mathematics, the number score is very young, and before the 19th century, it was not even a branch. The first mathematicians to realize the need to inject rigor into their analysis were Gauss and Abel, who quarreled over it. After a heated debate, Abel fell ill 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 angry with the young genius Abel of his time, and the grandmaster Gauss of the generation was responsible. ”

Gauss lived until he was almost 80 years old, and he wrote a treatise called Calculus in his later years, which we can consider to be the prototype of the number of points, in the middle of the 19th century. So again, the collision of ideas generates the impetus for academic development. At this point, Professor Lu paused.

All the students in the audience listened with relish, and sure enough, Gauss was still powerful, and he was angry with his academic theory, which is a powerful combat power that only a grandmaster has.

Perhaps the authenticity of Professor Lu's 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.

After a lecture on the history of mathematics and mobilizing the atmosphere of the classroom, Professor Lu freely entered the topic: "Standing on the shoulders of giants, after the further refinement of Cauchy and Weierstrass, in the early 20th century, Lebeguer completed the final work, and "Mathematical Analysis" became a worldwide mathematics course, which was arranged into the basic textbooks of mathematics departments of universities around the world. In the next few lessons, I will talk about Lebegus points, and Lebegus also has a lot of interesting stories to tell. ”

From the answers and discussions of the questions just now, we find that between the 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 a definite value neighborhood of variable x, and a fundamental property of a continuous function is not enough to ensure the continuity of the function. ”

"Dear students, please remember this basic nature, which arose from the collision of ideas of young mathematicians such as Shen Qi, Shao Tiantian, and Zhou Yu'an...... I hope you will become real mathematicians in the future. Professor Lu smiled.

Shen Qi, Shao Tiantian, and Zhou Yu'an also laughed, and they were encouraged, and the relationship between teachers and students tended to be harmonious between talking and laughing.

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

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

At the beginning of the lesson, some students were very repulsive to Professor Lu's style of asking questions without saying a word.

And now, everyone is waiting for the new question with great interest, and they are eager to try.

Professor Lu moisturized things silently, and used less than a class to make students go from rejection to acceptance of him.

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

"This time it's the math department's turn again. Professor Lu looked at Shen Qi, and he understood that Shen Qi was the core figure and boss of the Department of Mathematics. It seems that Shen Qi has a few fierce generals under him, and the boss generally does not go out easily, and if there is a problem, he will send the younger brother to solve it first, and the younger brother will not be able to solve it before it is the boss's turn to come forward.

Shen Qi looked back at the positions of Zhou Yu'an and Ou Ye, and gave Ou Ye a look: Calculate Ji, it's your turn this time.

Professor Lu followed Shen Qi's gaze and glanced at the back seat and locked on Ou Ye: "The first few are all boys to solve the problem, and then we will invite a girl to the stage, Ou Ye, please come to the stage." ”

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

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

"Okay, Ouye, what kind of thinking did you calculate this result on?" Professor Lu asked.

Oye replied, "Green's formula." ”

Professor Lu asked: "To be specific, I need details, more details." ”

Ou Ye looked at Shen Qi helplessly and did not speak.

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

Shen Qi stood up to solve the problem: "D is a closed curve surrounded by L and L1, and a square of the value e minus 1 can be calculated, and then by Green's formula, the final result is that I is equal to 1 minus e squared. This is my understanding of Ouye's thinking. ”

Professor Lu asked Ou Ye, "You think so too?"

Ouye nodded.

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

Ouye: "I will count, I won't tell." ”

Some students in the audience laughed, this girl is a little interesting, her calculations are very sharp, and her speech is not good.

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

"Alright, last question. ”

Professor Lu wiped the blackboard clean, drew a curve diagram, 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 left to the Department of Science and Engineering Computing. Professor Lu looked at Shao Tiantian.

This time, Shao Tiantian did not come to the stage immediately, he encountered confusion, he had no idea, and he did not know how to prove it.

No one from the Department of Science and Engineering Computing stepped forward, pretending to be ** is very easy, pretending to be big and relying on top strength, and no strength can only stare dryly.

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

Shen Qi stood up, this time he didn't send his little brother and sister out, he knew that he was the only one who could make a complete proof in the entire mathematics department of this problem. If there is a second one, it is Ou Ye, but the derivation and proof of this problem will be very cumbersome, and in Ou Ye's language expression style, she can't finish proving the idea for three days and three nights.

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

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

"PR and P1R1 are the tangents of the curves at the points of P and P1 respectively, so I will make the difference between the two definite integrals ......" Shen Qi said as he wrote, and wrote as he spoke.

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

......

"In the case of ellipses, I use algebraic expressions to express the difference of infinite segments of arcs, so the analysis is as follows......"

∫Xdx+∫Zdz=-hxz/√【-fl】

......

"It's a lot of trouble to prove, and let me think about it. Shen Qi wrote half of the blackboard and paused slightly.

In the audience, outstanding students such as Shao Tiantian and Zhou Yu'an, who were praised as "young mathematicians" by Professor Lu, were also dumbfounded, and they couldn't understand Shen Qi's idea of derivation and proof.

Professor Lu remained silent and kept watching.

"I thought of it, and here I am quoting geometry, so that this equation is consistent with the integral, and p is the sine of the ellipse......"

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

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

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

"So, I'll order again...... Hey, there's no land. Shen Qi wrote and found that a whole blackboard was filled with him, and there was no room for it.

Shen Qi turned around and threw half of the chalk into the blackboard slot: "I'm pretty sure this equation is true, but there are too few blank spaces on the blackboard to write." ”

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

"I'm pretty sure this assumption is true, but there are too few blank spaces in the book to write it down. That's how Fermat's theorem came about, and it wasn't until 1995 that Wiles proved it to be true.