In the afternoon, Chen Zhou's cousin Chen Yong came over with a schoolbag.

Chen Zhou arranged him and Chen Xiao together, let them write their own homework, and ask him if they didn't understand.

Very easily, Chen Zhou threw one of Chen Yong's math textbooks to Chen Xiao.

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

Chen Zhou looked at the two of them for a while, then went back to the house and took out his notebook, scratch paper and other equipment.

Open the notebook file on the topic of Clifford's analysis.

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

After briefly sorting out his thoughts, Chen Zhou began to write on the scratch 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 important equations that need to be proven first.

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

【∑j=0→n[(?? w1*/?? ξj+?? w2*/?? ξj)ej]=......】

[Obviously, the sum of these two corresponds is zero, and so on for the rest...... Therefore, the above formula is established. 】

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

After the proof was completed, Chen Zhou wrote down the next content that needed to be proved.

[Set Ω?? C^(n+1) is the bounded region, let f,g∈C1(Ω,Cl0,n(C)), define df=?? f+▔?? f，...... , then there is d[f?? (w1+w2)] =df∧(w1+w2)。 】

After a little thought, Chen Zhou began to prove.

[Because d(f?? g)=df?? g+f?? dg, so d[f?? (w1+w2)] =df∧(w1+w2)+f?? d(w1+w2)=df∧(w1+w2)+f[?? (w1+w2)+▔?? (w1+w2)] 】

[Because ▔?? w2=0，?? w1=0, so ......]

As soon as Chen Zhou finished writing, Chen Yong next to him poked him: "Brother, help me look at this question, I can't do this question, and I don't understand it after reading the answer." ”

Chen Zhou took the information book in his hand, glanced at it, a function of the problem, he raised his hand and wrote a?? and then immediately crossed out.

Shaking his head slightly, Chen Zhou muttered to himself, this is really a matter of what it is.

After reading the question again and sorting out his thoughts a little, Chen Zhou began to write the steps to solve the problem on the scratch paper while explaining to Chen Yong.

After stopping his pen, Chen Zhou glanced at Chen Yong, who was still staring at the scratch paper.

This question is indeed a bit out of the curriculum for high school students.

Chen Zhou was not in a hurry, just thinking about his own topic and waiting for Chen Yong.

After a while, Chen Yong withdrew his gaze from the scratch paper and turned his head 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 to do the question." ”

After speaking, Chen Zhou also returned to his own topic.

Now that the first two foreshadowing theorems have been settled, the following is a proof of the Cauchy-Pompieu formula.

The formulation of the Cauchy-Pompieu formula is:

[Set Ω?? C^(n+1) is the bounded region, let f∈C1(Ω,Cl0,n(C)), and f∈H(Ω,α)(0<α<1), then for any n+1-dimensional chain Γ, ▔Γ?? Ω, there is f(z)=∫?? Γf(ξ)?? (w1+w2)-∫Γd[f(ξ)?? (w1+w2)]。 】

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

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

According to Stokes' formula in the multiplex analysis, we can continue to prove it below.

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

After writing it, Chen Zhou looked back at it, mainly using the definition of the limit, and separating the part containing the singularity by digging points.

The part containing the singularity can be proved to have a limit of zero using the Herder's definition of the continuity of the function.

For the part without a singularity, the Stokes formula is used to prove that the result is a definite constant, so that the problem will be solved.

This afternoon, Chen Zhou spent his time in rotation among topics and explanations.

In the evening, he and Yang Yiyi opened the video again, supervised each other, and learned from each other.

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

The next day, Chen Zhou still spent the same time.

Except for the occasional question asked by Chen Xiao and Chen Yong, Chen Zhou took a brief break, and the rest of the time, he was immersed in the topic.

In the progress of the project, Chen Zhou has advanced to the study of the properties of the T operator with B-M nuclei in the complex Clifford analysis.

The relevant preparatory knowledge and definitions, Chen Zhou has already sorted out almost.

Like Hadamard's lemma, Herder's inequality, Minkowski's inequality, and so on, he already knows it by heart.

The T operator, the full name of which is Teodorescu operator, is a singular integral operator, which has many excellent properties and can be applied and studied in the theory of partial differential equations, integral equations and generalized functions.

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

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

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

Chen Zhou, who is concentrating on the research of the subject, only feels that time flies quickly.

Feeling that he hadn't done much content yet, Yang Yiyi reminded him that it was time to go to bed......

February 14, Valentine's Day.

According to the results of Chen Zhou and Yang Yiyi's discussion, neither of them plans to go out to meet, eat, watch movies or anything like that.

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

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

Therefore, Chen Zhou on this day was as usual, brushing books with Yang Yiyi in the morning to do the project.

In the afternoon, tutor Chen Xiao and Chen Yong.

Chen Xiao and Chen Yong glanced at each other, and Chen Xiao spoke 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 everyone else goes out on a date on Valentine's Day, and they are all one-on-one on the street, but you have been staying at home." ”

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

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

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

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

However, after Chen Xiao's reminder, Chen Zhou felt that there was some truth.

It's just that where is he going to prepare the gift now, and it's too late to prepare the gift now......