Train of Thought
Shen Qi volunteered to go up on stage and solve the problem.
He had studied mathematical analysis, from Mathematical Analysis I to Mathematical Analysis III, before coming to Yenching University.
Not to mention the basic courses in the Mathematics Department majors in the first year, such as mathematical analysis, Advanced Algebra, and Analytic Geometry, Shen Qi also taught himself subsequent professional courses such as Complex Analysis, Partial Differential Equations, and topology.
University is not high school. Here, the sky's the limit. Those who are capable should do more. If you have the ability and energy, then learn more. No one will stop you from learning.
The professors may have their own plans for placing a few outstanding students like Shen Qi in large classrooms to receive basic education alongside other students. Sun Erxiong told Shen Qi that, at least during his first year, he should attend classes in large classrooms.
The mathematical analysis problem that Professor Lu wrote on the blackboard was a bit too much. He was making things difficult for the first-year students. This problem had already gone beyond the scope of basic education, and today's mathematical analysis class was only Professor Lu's second class.
In universities, especially top institutions like Yenching University and Tsinghua University, the dozens or hundreds of class hours alone are not enough. Without putting in hard work outside of class, you should be grateful if you can avoid failing any courses, let alone achieve anything outstanding.
"This problem is difficult to solve using a unified projection, so we need to use a faceted projection." Shen Qi reorganized his thoughts and wrote his solution on the blackboard.
The equation of S is x^2 + y^2 = 4, which is not a continuous function like z = z(x, y), making it difficult to find the normal vector on the side where S is located.
If we use a faceted projection, the projection of the cylindrical surface onto the XOY plane is a line, or more precisely, a circle. Therefore, (z+l)dxdy=0
Next, Shen Qi began calculating the value of -ydzdx.
Determining the range of values for x and z requires drawing a graph. Shen Qi drew a planar projection graph on the blackboard and finally calculated I = -8π.
Shen Qi put the chalk back in the trough, dusted his hands off, and said to Professor Lu, "Solution complete."
Professor Lu: "Shen Qi, you have provided an answer. I will not judge your answer for now. Please explain your thought process."
"Okay." Shen Qi remained on the podium; his seat was the professor's seat.
Professor Lu stood to one side, with his hands behind his back.
Knock knock.
Shen Qi tapped the blackboard. "Y is an odd function that is symmetric to the cylindrical surface x² + y² = 4 about the plane XOZ. The 'S before' I wrote here refers to the part of the cylindrical surface x² + y² = 4 where y is greater than 0. So y equals 4 minus the square root of x². Do you all understand now?"
"Hmm, that makes sense." Zhou Yu'an nodded.
"That's what I think too," Luo Ji agreed.
"Shen Qi's approach is okay." Cai Qing scribbled in her notebook.
Ou Ye looked at him without saying anything. Her calculation result was exactly the same as Shen Qi's, but the calculation method was different.
These people were on the same wavelength as Shen Qi. While Shen Qi was explaining the solution on the blackboard, they also arrived at their own answers.
The other students in the Mathematics Department were a step behind, but after Shen Qi wrote out his solution, they could roughly understand it. After all, more than half of the students in the Mathematics Department were admitted to Yenching University through mathematics competitions, so their mathematical foundation was relatively solid. When they participated in the Golden Autumn Camp last year, they had already received education from Professor Sun Erxiong and undergraduate courses in the Mathematics Department.
From last autumn to this autumn, over the course of a year, Shen Qi was not the only one who enriched himself.
Sun Erxiong provided directional advice to each participant in the Golden Autumn Camp.
A year is enough time to learn a lot on your own, if you really focus on studying.
"Shen Qi, pause for a moment." Professor Lu interrupted Shen Qi's line of reasoning and called on a student, "Zhou Yu'an, you analyze this. Is what Shen Qi said correct? You don't need to stand up; just sit down and speak."
Zhou Yu'an replied confidently, "Shen Qi is absolutely right."
"Don't be so vague, I need details." Professor Lu scrutinized Zhou Yu'an.
Zhou Yu'an spoke eloquently, "Actually, Shen Qi's previous derivations and calculations were all standard procedures. The highlight was the diagram he drew. Drawing solid diagrams of surface integrals of the second kind is quite complicated, but Shen Qi simplified it by drawing the projection onto a certain plane, determining the range of values for x and z, and finally calculating that I equals -8π. I give Shen Qi 99 points for this, and I'm not giving him the remaining one point because I don't know if Shen Qi's answer is correct; that needs Professor Lu's judgment."
Professor Lu then asked Shen Qi, "Shen Qi, what are your thoughts on Zhou Yu'an's comments?"
Shen Qi placed his hands on the podium and looked at Zhou Yu'an: "Zhou Yu'an, regarding your comments about me, I'll give you 99 points for that. The remaining point is because you talked too much nonsense."
"Tch." Zhou Yu'an laughed and made a "come and hit me if you disagree" gesture.
The classroom atmosphere was quite lively; Professor Lu deliberately created this atmosphere. Even math professors admit that math is a dry subject. If a dry subject is taught in a dull way, how many students will genuinely like it?
"Shen Qi's answer and Zhou Yu'an's comments are both correct. Shen Qi's answer is correct, that's right, it's -8π," Professor Lu said, then turned to Shen Qi and said, "Shen Qi, you can return to your position now. Please give me back my podium."
Shen Qi stepped down from the podium and returned to his seat.
Professor Lu returned to his position at the center of the podium. He tapped the blackboard with the eraser on the projection of Shen Qi's drawing. "This is important; you need to take notes."
The Mathematics Department students in the audience took out their pens and notebooks, ready to take notes.
Most students in the Department of Scientific and Engineering Computing looked bewildered. "Important? What's important? Where is it? Why is it suddenly important? Ah, where am I? Who am I? Where am I going...?"
There's no harm without comparison.
The overall foundation of the Mathematics Department students is significantly stronger than that of students in the science and engineering computing department.
In fact, mathematical analysis is a very important course in the Department of Scientific and Engineering Computing. It is almost a purely applied course; it is a mathematical computational tool. Perhaps because it's the beginning of the semester, the students in the Department of Scientific and Engineering Computing haven't quite gotten into the swing of things yet.
Professor Lu gave the students a problem to solve, asked them to explain their thought process, and then had other students provide feedback.
If students teach students, then what's the point of having professors?
The professor, of course, was the one who added the finishing touch. Professor Lu summarized. "I already mentioned this in the last class, and now I'll emphasize it again: the symmetry of surface integrals of the second kind. Its pattern is exactly the opposite of the pattern of odd functions of double and triple integrals. Does everyone understand?"
"If you don't quite understand, you should spend more time figuring it out on your own after class. Most of you have received advance training such as the Yenching University Mathematics Golden Autumn Camp and Winter Camp. Don't tell me you've only started learning about mathematical analysis these past few days."
Students who didn't participate in the Yenching University training camp were speechless with frustration. Their numbers were small, and a small portion of them were admitted to the School of Mathematics at Yenching Big Mathematics Department through the national college entrance examination.
This small group of students can be divided into two categories: one group voluntarily applied to Yenching University's mathematics program when filling out their college applications, and the other group scored very high on the college entrance exam, listed Yenching University as their first choice but did not select mathematics as their major. They don't know why they ended up in Yenching Big Mathematics Department. It's about half and half of the two groups.
Professors sometimes have to make choices: should they cater to the majority or favor a small minority?
Professor Lu's approach is to cater to the majority.
Therefore, those students who didn't quite understand the lesson had to work even harder to catch up, otherwise they would fall behind and become increasingly unable to keep up with the main group's pace as time went on.