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Chapter 255 Differential Equations, Conjugate Gradients, Taylor's Formula!

On the A4 paper size paper, there are three questions listed.

All three questions have traces of being circled.

Naturally, Professor Lu would not have known in advance that Cheng Nuo was going to come to him to apply for exemption.

So……

The topic he seemingly randomly picked out from the stack of materials on his desk.It was not specially prepared for Cheng Nuo.

Judging from the traces of the circle on the paper, these three questions have been done once before.

And that person is most likely Professor Lu sitting in front of him.

However, figuring this out is of no use to Cheng Nuo's current situation.

No matter how these three questions came about, and who had done them before, if Cheng Nuo wanted Professor Lu to sign the exemption application form, he had to do one of these three questions.

Choose one of the three and get it right!
With Professor Lu's character, being able to put forward such a condition is enough to prove that the three questions on the piece of paper Cheng Nuo is holding in his hand are definitely not for nothing!
Its power can definitely kill tens of thousands of scumbags in an instant!
There was no room for Cheng Nuo to be careless.

Cheng Nuo looked at Professor Lu who was sitting at the desk, stepped forward and said, "Teacher, I didn't bring my schoolbag, can I borrow a pen and draft paper?"

Professor Lu put down his pen, looked up at Cheng Nuo who had a harmless smile on his face, bent down, opened the desk drawer, and handed the pen and draft paper to Cheng Nuo.

He pointed to a desk beside him, "Just do it over there, and call me when you're done."

After speaking, he lowered his head again and continued the work in his hand.

And Cheng Nuo was obedient, took a pen and draft paper, walked to the desk Professor Lu pointed out, pulled a chair and sat down.

The A4 paper with three questions listed was also laid flat on the table by Cheng Nuo.

Cheng Nuo looked at the three questions in turn and decided which one to choose as a breakthrough.

The first question: [The known elliptical cylinder S.

r(u, v)={acosu, bsinu, v}, -π≤u≤π, ﹣∞≤v≤+∞

(1): Find the equation of any geodesic on S.

(2): Let a=b, take p=(a, 0, 0), Q=r(u, v)={acosu0, bsinu0, v0}, -π≤u0≤π, -∞≤v0≤+ ∞, write down the shortest curve equation on S connecting two points P and Q. 】

The second problem: [Deduce the calculation format of the conjugate gradient method for solving linear equations, and prove that the format converges after a finite number of iterations. 】

第三题：【设f(x)在[0，1]上二阶可导，且f(0)=f(1)=0，min（0≤x≤1)f(x)=-1。

Proof: There exists η∈(0) such that f(η)>1. 】

After reading the three questions from beginning to end, Cheng Nuo frowned.

The first question is a very comprehensive one.

Elliptic equations, trigonometric functions, differential equations, vector operations.

The combination of the four aspects makes this question extremely difficult.

Solving the first question requires knowledge of vectors and trigonometric functions, which is not difficult for Cheng Nuo.

But the second question mainly requires the knowledge of ordinary differential equations.

Regarding ordinary differential equations, it is actually covered in the last chapter of the volume [-] of "Advanced Mathematics" that Professor Lu is teaching.

However, it is originally a basic mathematics teaching book, and the content of advanced mathematics is just some of the most basic and simple solutions, just superficial.

Even, perhaps not even the fur.

In the Department of Mathematics, when I was a sophomore, there was only a professional course called "Ordinary Differential Equations", which explained such equations in detail.Cheng Nuo took classes with this year's freshman in the mathematics department, so naturally he hasn't learned it yet.

Judging from Cheng Nuo's only knowledge, the second question should be solved by using the Picard-Lindelof theorem for ordinary differential equations.

But Cheng Nuo had only heard a little about the Pica-Lindelof theorem.Cheng Nuo still had a long way to go to use the distance flexibly.

For the first question, Cheng Nuo could only give up strategically.

As for the second question, it made Cheng Nuo even more painful.

The so-called conjugate gradient method of a system of linear equations obtains a large system of linear equations by differentially discretizing the Laplace equation.

The requirement of the title is to carry out continuous iterative operation on the general format of this equation system, and determine the orthogonal equation system through the recursive relationship of residuals, and determine the convergent value of that approach.

To say that the method of solving the differential equation in the first question is barely related to advanced mathematics.

The second question has nothing to do with the content explained in the high mathematics course!
The conjugate gradient method, the Laplace equation, and the recurrence relationship of residuals are not at all what Cheng Nuo, a freshman, should master.

And indeed, like the previous question, Cheng Nuo had only heard about the content.

As for solving the problem, I'm sorry, Cheng Nuo really can't do it!
Originally, Cheng Nuo thought he would solve all three of these questions, which shocked Professor Lu well.

However... not enough strength.

However, Cheng Nuo was fortunate that the third question was very friendly to Cheng Nuo.As long as the special form of Taylor's formula is used, McLaughlin's expansion, and the relevant knowledge of the Schlemilch-Rosch remainder, it can be solved perfectly.

Taylor's formula is considered to be the most complicated and difficult content in the entire advanced mathematics volume [-] knowledge.Countless arrogance was buried here.

It is generally used to calculate errors.For general questions about Taylor's formula, only a simple formula is required.

But the question in front of Cheng Nuo was not like that.

That really needs to be expanded one by one with Taylor's formula.

The workload is quite complicated!

But compared to the first two questions, which were completely incomprehensible, Cheng Nuo could only choose this question that tested the amount of calculation.

Let's get to work!

Cheng Nuo rubbed his hands and brought a stack of draft paper in front of him.

Now that you have chosen a topic, do your best to do it.

The exemption application, I must get it!
With his eyes closed, thoughts raced through his mind at high speed.

Half a minute later, Cheng Nuo's eyes suddenly opened, and a flash of light flashed across them.The corners of his mouth were slightly raised, he picked up the pen, and calculated while writing on the draft paper.

【f(x)=f(t)/0!+f'(t)/1!*(xa)+f''(t)/2!*(xa)^2...

…………

0=f(0)=-1+f''(t1)/2!x0^2
0=f(1)=...

又因为0≤x≤1，所以f(η)=max{2/x^2，2/(1-x0)^2}≥8 ！】

Get it!

After more than ten minutes, Cheng Nuo listed the formulas on a whole sheet of A4 paper, and finally figured out the problem.

At that moment, I felt a sense of accomplishment.

After checking and confirming that there were no questions, Cheng Nuo capped the pen, picked up his answer, got up and walked to Professor Lu.

"Professor, I'm done." Cheng Nuo said softly.

Professor Lu looked up at Cheng Nuo first, then raised his wrist to check the time.

There was also a slightly surprised expression on his slightly serious face.

Obviously, Cheng Nuo's speed exceeded his expectations.

He looked up and down seriously, but he was not in a hurry to answer Cheng Nuo's written answer, but asked with a smile, "What question did you do?"

"The third way." Cheng Nuo answered honestly.

"Then do you know where I got these three questions?" Professor Lu asked.

Cheng Nuo shook his head.

Professor Lu, please spit out a sentence, "The last three questions in the third and fourth grade finals of the national college mathematics competition last year were these three questions."

"That time, none of the students could answer all the last three questions correctly."

(End of this chapter)