"Student Li Mo, your application for early graduation has been received. After research by the Department's Academic Affairs Office, it has been decided to take the exam for you on Wednesday and Thursday in Lecture Hall No. 5. I hope you can attend on time."

Li Mo breathed a sigh of relief when he saw the letter in his mailbox. After passing the exam, he could graduate directly and participate in the research project in advance.

Although Li Mo was very confident that he would pass the exam, he still went to the corner of the library to review all the subjects again.

"Li Mo, have you decided to graduate early?"

Li Mo turned around and saw Ying Sasha standing behind him with a slight frown.

"Yes, the Academic Affairs Office has approved my application and the exam will be held this week." Li Mo said with a calm expression.

The corners of Ying Sa Sa's eyes turned red when she heard this, "Then are you planning to study abroad? Can we meet again?"

Li Mo looked slightly stunned, with a hint of smile in the corner of his eyes, "I am already preparing to apply for Professor Wu's graduate school, and will continue to stay at Yanda."

"Really, that's great." Ying Sa Sa was overjoyed, perhaps realizing her gaffe, she said a little embarrassedly: "Then you continue reading, and I will go and deliver a book to them."

Li Mo thought for a long time as he watched Yingshasa's back leaving with light steps.

WeChat

Zhou Dayong: "The installation workers have already entered the site, and Brother Qiu and I are watching at the scene."

Li Mo thought for a moment and replied: "Thank you, please install the workers according to the drawings."

Zhou Dayong: "Received!"

"There are two bodyguards supervising the work on site, so maybe I can relax." Li Mo felt that he was a bit weak. As the saying goes: "You are covered with iron, how many nails can you drive?".

Wednesday morning, Lecture Hall 5.

Li Mo came to the classroom on time. In the empty classroom there was only a teacher from the Academic Affairs Office with a test paper and a graduate student who was temporarily arrested.

"Junior brother Li Mo, I am Professor Wu's graduate student, Zhou Ming." The boy wearing glasses shyly greeted him.

Li Mo smiled calmly and said, "What a coincidence. I'm about to apply for Professor Wu's graduate program. Maybe you will be my senior brother in the future. Please take good care of me."

Zhou Ming said modestly: "Your name has spread throughout Yanda for a long time. Professor Wu always praised you in front of us as a rare mathematical genius."

"I am over-flattered."

"Cough, cough." The teacher from the Academic Affairs Office next to him saw that the two of them were engaged in household chores and interrupted, "Since time is tight, let's start the exam now, classmate Li Mo."

"The exams scheduled for this morning are three subjects, "Mathematical Analysis," "Advanced Algebra," and "Calculus Equations." Since it is an early exam, it will not be held according to the normal exam time."

"Before 12 noon, you just need to hand over the three completed test papers to me." He handed out the three test papers.

The first test paper is "Mathematical Analysis",

1. For the leaf-shaped line x=2t-t?, y=2t?-t?, 0≤t≤2, find the area of ​​the figure enclosed by this curve.

This is too simple. Li Mo came up with the answer after a little thought. He wrote on the test paper:

|y=tx,t00.511.52x00.7510.750y00.37511.1250, area a=∫<0,1>(2t-t^41022)(2-2t)dt=∫<0,1>(4t-6t^22t

^3)dt=(2t^2-2t^3t^4/2)|<0,1>=1/2.

2. All partial derivatives of u=(x/y)^(1/z) at (1,1,1).

This question was not difficult for him. In less than 2 seconds, Li Mo deduced the answer:

u=u(x,y,z)?u/?x=[(x/y)^5261(1/z)]/(zx)=u/(zx)?u/?y=-[(x

/y)^(1/z)]/(zy)=-u/(zy)?u/?z=-[(x/y)^(1/z)](1/z?)ln(x

/y)=-u[ln(x/y)]/z?u=(x/y)^(1/z)在(1,41021,1)1653u=u(1,1,1)=1

?u/?x=1,?u/?y=-1,?u/?z=0

3. Find the total differential of u=ln(sin(xy))

1 second, only 1 second, Li Mo wrote down the answer directly.

du=(?u/?x)dx(?u/?y)dy?u/?x=y[cos(xy)]/[sin(xy)]?u/?y=x[cos(xy)

]/[sin(xy)]du=(ydxxdy)[cos(xy)]/[sin(xy)]

............................

.............................

It only took Li Mo 30 minutes to finish the "Mathematical Analysis" test paper. If it weren't for the last open-ended question, which he used 6 methods to explain, it could have been faster.

The next test paper is "Advanced Algebra".

1. Assume that v1 and v2 are the solution spaces of the homogeneous equations x1x2...xn=0 and x1=x2=...=xn respectively. Find v1 and v2 and prove that p^n=v1v2, where

p^n is an n-dimensional vector space on the number field p.

Answer: v1 is the orthogonal complement space of vector bai(1,1,...,1), with the bases (1,-1,0,0,...,0), (du1,0,-zhi1,

0,...,0),...,(1,0,...,-1), the first component of each vector is 1, the k1th component is -1, and the remaining components are 0,k

=1,2,...,n-1. The basis of v2 is (1,1,1,...,1). It is easy to see that v1 and v2 are orthogonal (the basis vectors are orthogonal

), the dimension of v1 is n-1, the dimension of v2 is 1, and the sum of the two is n, so the sum of the two subspaces is a direct sum, which happens to be the whole space.

He completed the first question in 1 minute. Since his spiritual intelligence reached level 2, he felt that he could easily grasp the ideas for solving the problem.

Zhou Ming, who was next to him, saw that Li Mo had completed the "Mathematical Analysis" test paper, and couldn't help but walk behind him and start reading. The young boy in front of him was writing the questions as fast as he was writing an article.

Even if he encounters a cunning question, the young man will frown slightly and think about it for a while, and he will be able to solve it easily.

Professor Wu often boasted in front of him that a genius had emerged from the Mathematics Department. Zhou Ming did not believe it at first. Anyone who could enter the Mathematics Department of Yanda was not a genius.

But now when he saw the young man in front of him who was doing the questions quickly, Zhou Ming truly understood the meaning of genius.

The "Advanced Algebra" test paper was also completed quickly by Li Mo. Zhou Ming subconsciously checked his mobile phone and it only took 20 minutes.

The next test paper is "Calculus Equations". "Calculus Equations" is famous for its large amount of calculations. It is not the kind of question that can be easily solved if you have ideas for solving the problem.

"How long will it take to see you this time?" Zhou Ming specifically checked his mobile phone this time. It was now 9:30.

The first question is, assuming a, b, c are all positive constants, and y(x) is a solution of the differential equation ay‘by‘cy=0, prove:

lim(n~∞)y(x)=0.

Li Mo quickly calculated it in his mind and wrote:

ar^2brc=0

Because a,b,c>0

The two roots are r1 and r2

By Wade's theorem

r1r2=-b/a<0

r1*r2=c/a>0

If r1 and r2 are real roots, then obviously only r1 and r2<0 can satisfy that the sum is less than zero and the product is greater than zero.

.......

When x->positive infinity, exp(r1*x), exp(r2*x)->0, so y->0

If it is a complex root, it must be a conjugate complex root because the coefficients are real numbers

So r1=min,r2=m-in

r1r2=2m=-b/a<0

.......

Because m<0

exp(mx)->0 when x->positive infinity

|c1cosnxc2sinnx|<=|c1||c2|bounded

So when x->positive infinity y->0

To sum up, lim(n->∞)y(x)=0

It was so long, and my hands were so tired. Calculus equations questions are indeed notoriously complicated. Li Mo worked hard.

Lane 2... Lane 3... Lane 4...

It was finally over, and Li Mo wrote the last mathematical symbol on the test paper.

Zhou Ming checked his phone again. At 9:55 and in 25 minutes, the young man in front of him completed this set of test papers that took him at least 2 hours to complete.

"Teacher, I've finished, can I hand in the paper in advance?" Li Mo asked, raising his hand.

"Did you finish?" The teacher from the Academic Affairs Office who was in a daze at the door was a little surprised. There were three test papers. It was originally scheduled to be completed in one morning and it was already meant to test him.

I didn't expect him to finish it so quickly.
Chapter completed!