Xueba starts with change

Chapter 157 Advancing the subject

In the afternoon, Chen Zhou's cousin Chen Yong came over with his schoolbag on his back.

Chen Zhou arranged him and Chen Xiao together, let them do homework by themselves, and asked him if they didn't understand.

Very smoothly, Chen Zhou threw one of Chen Yong's mathematics textbooks to Chen Xiao.

Chen Xiao took it silently. He knew that this winter vacation, this textbook would always be with him.

Chen Zhou watched the two of them for a while, then went back to the room and took out his notebook, draft paper and other equipment.

Open the file in the notebook on the topic related to the Clifford analysis.

He is now working on the relevant part of the Cauchy-Pompeiu formula in complex Clifford analysis.

After briefly sorting out his thoughts, Chen Zhou began to write on the draft paper:

【w1*Dξ+w2*Dξ=∑j=0→n[(w1*/ξj+w2*/ξj)ej]=0……（1）】

【Dξw1*+Dξw2*=∑j=0→n[ej(w1*/ξj+w2*/ξj)]=0……（2）】

These two are very important equations that need to be proved first.

Chen Zhou thought for a while, made some changes to the above two equations, and then proceeded to prove it.

【∑j=0→n[(w1*/ξj+w2*/ξj)ej]=……】

[Obviously, the sum of these two corresponding items is zero, and the rest of the items can be deduced in this way... so the above formula is established. 】

[The same reason can prove that Dξw1*+Dξw2*=0]

After the proof was completed, Chen Zhou wrote down the next thing to be proved.

[Assume ΩC^(n+1) is a bounded area, let f, g∈C1(Ω, Cl0, n(C)), define df=f+▔f,..., then d[f(w1+w2 )]=df∧(w1+w2). 】

After a little thought, Chen Zhou began to prove.

【因为d(fg)=dfg+fdg，所以d[f(w1+w2)]=df∧(w1+w2)+fd(w1+w2)=df∧(w1+w2)+f[(w1+w2)+▔(w1+w2)]】

【Because ▔w2=0, w1=0, so...】

Just as Chen Zhou finished writing, Chen Yong next to him poked him: "Brother, help me read this question, I don't know how to do this question, and I didn't understand the answer."

Chen Zhou took the information book in his hand and glanced at it. He raised his hand to write a symbol on a function title, and then crossed it out immediately.

Shaking his head slightly, Chen Zhou muttered to himself, it really depends on what it is.

After reading the question again and sorting out his thoughts a bit, Chen Zhou began to write the problem-solving steps on the draft paper and explain it to Chen Yong.

After he stopped writing, Chen Zhou glanced at Chen Yong, who was still staring at the draft paper.

This question is indeed a bit beyond the outline for high school students.

Chen Zhou was not in a hurry, he just waited for Chen Yong while thinking about his subject.

After a while, Chen Yong withdrew his gaze from the draft paper and turned to look at Chen Zhou.

Chen Zhou smiled and asked, "Do you understand?"

Chen Yong nodded: "Well, thank you brother."

Chen Zhou: "You're welcome, let's continue with the questions."

After finishing speaking, Chen Zhou also returned to his subject.

The previous two foreshadowing theorems have been settled, and the following is the proof of the Cauchy-Pompieu formula.

The expression of the Cauchy-Pompieu formula is:

[Assume ΩC^(n+1) is a bounded area, let f∈C1(Ω, Cl0, n(C)), and f∈H(Ω, α) (0<α<1), then for any The n+1-dimensional chain Γ, ▔ΓΩ has f(z)=∫Γf(ξ)(w1+w2)-∫Γd[f(ξ)(w1+w2)]. 】

Chen Zhou took a pen, habitually clicked twice on the draft paper, and then began to prove.

[With z∈Ω as the center and sufficiently small ε as the radius, make a small ball Bε={ξ||ξ-z|<ε}, then...]

According to the Stokes formula in multiple analysis, the proof can be continued.

[..., when ε→0, ∫Bε[f(ξ)-f(z)](w1+w2)→0,...]

After finishing writing, Chen Zhou read it again, mainly using the definition of limit, and separated the part containing the singularity by digging points.

Among them, the part containing the singularity can use the definition of the Herder continuity of the function to prove that its limit is zero.

For the part without singularity, use the Stokes formula to prove that the result is a definite constant, thus solving the problem.

This afternoon, Chen Zhou spent his time alternating between topics and explanations.

In the evening, I will play video with Yang Yiyi, supervise and learn from each other.

It wasn't until Yang Yiyi urged Chen Zhou to go to bed quickly that he put down his pen and cleared his mind of thoughts.

The next day, Chen Zhou still spent like this.

Except for occasional questions being asked by Chen Xiao and Chen Yong, Chen Zhou took a short break, and spent the rest of the time immersed in the subject.

With the progress of the project, Chen Zhou has advanced to the research on the properties of the T operator with BM kernel in complex Clifford analysis.

Chen Zhou has sorted out the relevant preliminary knowledge and definitions a long time ago.

Like Hadamard's lemma, Herder's inequality, Minkowski's inequality, etc., he is already familiar with it.

T operator, the full name is Teodorescu operator, is a singular integral operator. This singular integral operator has many excellent properties and can be applied and studied in partial differential equation theory, integral equation theory and generalized function theory.

Looking at the conclusion he got, Chen Zhou thought of the conclusion of the classic Hile lemma, which is very similar.

However, because the Hile lemma cannot be directly used in the complex Clifford analysis, Zhou Chen inserted appropriate terms according to different situations to prove the relevant conclusions.

This conclusion is an important tool to prove the Herder continuity of operators in complex Clifford analysis.

Chen Zhou, who devoted himself to the research of the project, only felt that the time passed quickly.

Feeling that he hasn't done much yet, Yang Yiyi reminded him that it's time to sleep...

February 2, Valentine's Day.

According to the results of the discussion between Chen Zhou and Yang Yiyi, neither of them planned to go out to meet, eat, watch movies and so on.

After all, they had just separated, and they had been together since school, seeing each other every day, so there was no need to run out alone for the so-called Valentine's Day.

Generally speaking, both of them feel that as long as two people are together, every day is actually Valentine's Day.

So, today Chen Zhou was the same as usual, brushing books with Yang Yiyi in the morning to do a project.

In the afternoon, tutor Chen Xiao and Chen Yong.

Chen Xiao and Chen Yong looked at each other, and Chen Xiao said first, "Brother, did you break up with your sister-in-law?"

Chen Zhou asked strangely: "Why do you say that?"

Chen Xiao explained: "I see that other people go out on dates on Valentine's Day, and there are one-on-one couples on the street, but you have been staying at home all the time."

Chen Yong also said: "When I came, I also saw that there were flower sellers on the street."

Chen Zhou glanced at the two boys, and said helplessly, "You two really... I didn't break up, you two hurry up and do your homework."

But Chen Xiao said: "Brother, don't blame me for not reminding you, this necessary festival is still worth it. If you really haven't broken up, even if you don't see each other, you still have to prepare a gift for sister-in-law, right?"

Chen Zhou glared at Chen Xiao, who immediately lowered his head and didn't say a word.

However, after some reminders from Chen Xiao, Chen Zhou felt that there was some truth to it.

It's just that where is he going to prepare presents now? It's too late to prepare presents...

(End of this chapter)