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 and asked them to do their homework by themselves. If they didn't understand, they would ask him.

It was very easy to use, Chen Zhou threw Chen Yong's math textbook to Chen Xiao.

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

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

Open the file on your notebook about clifford analysis related topics.

What he is now studying is the relevant part of the cauchy-pompeiu formula in the complex clifford analysis.

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

【w1*dξw2*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 and need to be proved first.

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

【∑j=0→n[(?w1*/?ξj?w2*/?ξj)ej]=…】

[Obviously, the sum of these two corresponding terms is zero, and the rest are so on... Therefore, the above formula holds.]

【Similarly, dξw1*dξw2*=0】

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

[Suppose Ω?c^(n1) is a bounded area, let f, g∈c1(Ω,cl0,n(c)), define df=?f▔?f,..., then d[f?(

w1w2)]=df∧(w1w2).]

After a little thought, Chen Zhou began to prove it.

[Because d(f?g)=df?gf?dg, d[f?(w1w2)]=df∧(w1w2)f?d(w1w2)=df∧(w1w2)f[?(w1w2)▔?

(w1w2)]]

【Because ▔?w2=0,?w1=0, so…】

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

Chen Zhou took the information book in his hand and took a look at a function title. He raised his hand and wrote a symbol of "?, and then crossed it out immediately.

Shaking his head slightly, Chen Zhou muttered to himself, "It's really a matter of what it means."

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 draft paper while explaining it to Chen Yong.

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

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

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

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

Chen Zhou asked with a smile: "Do you understand?"

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

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

After saying that, Chen Zhou also returned to his own topic.

The first two theorems have been completed, and the following is the proof of the cauchy-pompieu formula.

The cauchy-pompieu formula is expressed as:

[Suppose Ω?c^(n1) is a bounded area, let f∈c1(Ω,cl0,n(c)), and f∈h(Ω,α)(0＜α＜1), then any

n1 dimensional chain Γ, ▔Γ?Ω, has f(z)=∫?Γf(ξ)?(w1w2)-∫Γd[f(ξ)?(w1w2)].]

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

【Take z∈Ω as the center, and the sufficiently small ε as the radius, make a small sphere bε={ξ||ξ-z|＜ε}, then...】

According to the Stokes formula in the multiple replication analysis, we can continue to prove it.

【…,When ε→0, ∫?bε[f(ξ)-f(z)](w1w2)→0,…]

After writing, Chen Zhou looked back, mainly using the definition of limits to separate the parts containing singularities by digging points.

Among them, the part containing the singularity can be used to prove that its limit is zero using the definition of Held's continuity of the function.

For parts without singularity, the Stokes formula is used to prove that the result is a definite constant, thereby solving the problem.

That afternoon, Chen Zhou spent the topic and explanation in a cycle.

At night, I would play videos with Yang Yiyi, supervise and learn from each other.

It was not until Yang Yiyi urged Chen Zhou to go to bed quickly that he put down his pen and cleared his thoughts in his mind.

The next day, Chen Zhou still spent this.

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

Chen Zhou has advanced to the study of the properties of the t operator with b-m core in complex clifford analysis.

Chen Zhou has already sorted out the relevant preliminary knowledge and definitions.

He is already familiar with the hasamard lemma, Held inequality, Minkowski inequality, etc.

The t operator, whose 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 classic Hile Lemma conclusion, which was very similar.

However, because the hile lemma cannot be used directly 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 operator Held in complex clifford analysis.

Chen Zhou, who was devoted to researching the subject, only felt that time passed quickly.

I feel like I haven't done much content yet, so Yang Yiyi reminded him again that it's time to go to bed...

February 14, Valentine's Day.

According to the discussion between Chen Zhou and Yang Yiyi, neither of them planned to go out to meet again, have dinner, watch movies, etc.

After all, I have just separated and have been together when I was in school. I meet every day. There is no need to go out alone for the so-called Valentine's Day.

In general, both of them think that as long as they are together, it is actually Valentine's Day every day.

So, Chen Zhou was the same as usual on this day, and he was working on a topic with Yang Yiyi in the morning.

In the afternoon, I tutored Chen Xiao and Chen Yong.

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

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

Chen Xiao explained: "I think others go out for dates on Valentine's Day, and they are all one-on-one on the streets, but you've been staying at home all the time."

Chen Yong also said: "When I came, I saw that there were flowers selling 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 well."

Chen Xiao said, "Brother, don't blame me for not reminding you that you still have to spend this necessary holiday. If you really don't break 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, and Chen Xiao immediately lowered his head and said nothing.

However, after Chen Xiao's reminder, Chen Zhou felt that there was a little reason.
Chapter completed!