Zhou Bigmouth's Skills
According to the syllabus for Mathematical Analysis, it should start with the limits of sequences and functions, and then move on to the areas of Leibniz and L'Hôpital, namely derivatives and differentials.
Then Taylor and Riemann came into the picture. Taylor series and Riemann integral will be used throughout mathematical analysis, and even throughout the entire career of mathematicians.
The above is the teaching content for Mathematical Analysis I.
This is how it should be under normal circumstances.
It was obvious that Professor Lu did not follow the usual procedure.
The calculation of double integrals, triple integrals, and n-fold integrals, as well as the concepts of surface integrals of the second kind and integrals in the form of external differentials on oriented manifolds, are the teaching content of Mathematical Analysis II.
In conclusion, Professor Lu did not simply copy the Ministry of Education's teaching syllabus; he had his own teaching plan. Furthermore, Yenching University was not under the jurisdiction of the Ministry of Education, but rather directly under the central government.
In his thirty years of research and teaching, Professor Lu has developed a set of effective teaching systems. He doesn't act haphazardly; he is a full professor at Yenching University and has received awards.
In Shen Qi's view, Professor Lu was at least at the level of a math expert with a level of 8 or higher; how could such a mathematical elite act recklessly?
Shen Qi liked Professor Lu's teaching style. Yes, that's right. There should be in-depth interaction between teachers and students, and among classmates. Vigorous collisions will inevitably produce extraordinary results, and exchanges that reach the soul will breed an explosion of academic achievements.
Professor Lu used questions as a starting point to help students better understand the calculation methods, theoretical concepts, and related techniques of mathematical analysis.
The first question involved calculating multiple integrals, and the two class leaders from the Mathematics Department and the Department of Scientific and Engineering Computing demonstrated their abilities, which Professor Lu acknowledged.
Professor Lu continued in his teaching style, posing another problem: calculate the limit.
lim[x→0](e^x+e^2x+... e^nx)1/x/n
"The Mathematics Department completed the solution to the first question, and the Department of Scientific and Engineering Computing supplemented it with new algorithms and approaches. Now, for the second question, please send someone from the Department of Scientific and Engineering Computing to come up and find the limit," Professor Lu said.
"I'll go," Shao Tiantian raised her hand.
"It's you again, Shao Tiantian? Don't the other students in the Department of Scientific and Engineering Computing have any thoughts?" Professor Lu asked.
"No ideas, no ideas, let him do it, let him do it." The other students recommended class monitor Shao Tiantian to go on stage to bring glory to the class and honor to the department.
"Alright, Shao Tiantian it is, you come." Professor Lu had no choice.
Shao Tiantian went on stage and quickly completed the calculation. He looked down at the audience with a smile, confident and assured.
"Please have the Mathematics Department students comment on Shao Tiantian's limit calculations." Professor Lu looked at Shen Qi's position.
The Mathematics Department was closely united around the central leadership with Comrade Shen Qi at its core. The students in their class sat together, except for Ou Ye and Zhou Yu'an in the back row.
"Let me go first," Shen Qi began. "Shao Tiantian's calculations are completely correct; my answer is the same as his."
"Oh." Professor Lu nodded, expecting Shen Qi to continue his analysis and commentary.
Shen Qi: "I'm done."
Professor Lu: "That's it?"
Shen Qi explained, "For things like limits, you just need to calculate the answer. The process is all standard, and I believe you students will understand it after looking at it, so I won't go into details."
"They can't even be bothered to give feedback, do they look down on our class monitor every day?" Students from the Department of Science and Engineering Computing had a lot of complaints about Shen Qi.
"I see. Then Shao Tiantian, you may leave. Your calculation is correct." Professor Lu respected Shen Qi's answer.
As Shao Tiantian stepped off the stage and passed Shen Qi's seat, he chuckled, "Hehe."
Shen Qi chuckled in response, "Hehehe." He added an extra "he" to his tone, putting him at a disadvantage.
The third question is again about calculating limits, specifically double limits.
This time it was the Mathematics Department's turn to send someone. Shen Qi turned to Zhou Yu'an and conveyed a message to Zhou Yu'an with his eyes: Zhou Yu'an, you should solve this problem. It's not appropriate for me to solve something as simple as limits.
Zhou Yu'an understood perfectly, and he went on stage to solve this double limit problem.
As is customary, before Zhou Yu'an went down, he had to explain his thought process and share his successful experience with the students of the School of Mathematical Sciences. Everyone would exchange ideas and discuss, correcting mistakes and striving for improvement together.
Zhou Yu'an shared a dorm room with Shen Qi. During their daily studies, Zhou Yu'an paid close attention to Shen Qi and secretly learned some mathematical techniques from him. He then eloquently explained, "Dear classmates, did you know that finding a double limit is much more difficult than finding the limit of a single-variable function? The difference between the two is like heaven and earth."
"Why? Because the domain of a univariate function is the set of points on the real axis, and finding the limit only requires considering the case of approaching from the left and right directions."
"The domain of a bivariate function is a set of points on the coordinate plane; in other words, finding a double limit requires considering the approach path from any direction..."
Professor Lu snapped, "Zhou Yu'an, get to the point! Everyone's time is precious, don't just recite textbooks!"
"Alright, alright, Professor Lu, don't worry, we're almost at the key point." Zhou Yu'an remained calm and composed, continuing slowly, "Classmates, do you know how many ways there are to calculate a double limit? I'm not very skilled, and I've only mastered five methods so far. The first one, using the definition to calculate a double limit, is the simplest but also the easiest to make mistakes on."
"The second method is to calculate the double limit using trigonometric substitution..."
"The third method is to use the continuity of the function to calculate the double limit..."
"The fourth method is to use the idea of important limits of univariate functions to find double limits. Here I want to explain this fourth algorithm in detail, which is also my core algorithm, and it is used to solve the double limit problem on the blackboard."
"First, I want to explain how to find double limits using the idea of important limits of a single-variable function. The core principle of this method is..."
Biu!
Professor Lu slammed his hand on the wall, making the blackboard shake. "Zhou Yu'an, I'll give you another 20 seconds! The double limit algorithm could be explained in just three sentences, but you've rambled on for a full five minutes! Zhou Yu'an, what's your zodiac sign?"
Sigh, Shen Qi looked out the window speechlessly. Zhou Yu'an was too damn slow. He was actually quite good at calculating this double limit. I taught him this method, and he showed off well. He had already mastered seven or eight parts of my teachings. At this moment, the most elegant thing to do was to explain the core idea in one sentence and then leave the stage with a swagger. That would also earn him some prestige.
But Zhou Yu'an, on the other hand, was so indecisive, like a woman. All his previous efforts were wasted, brother. He made her lose all her class and even looked a bit low-class. Most women in this world don't talk as much nonsense as you, Zhou Da Zui... Shen Qi clenched his fist and rested his chin on his hand, helplessly gazing out the window at the air.
At the same time, Ou Ye in the back row made a similar gesture to Shen Qi, resting her chin on her hand. Sigh, she sighed too, it was unclear what she was thinking.