I kidnapped an alien civilization

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 in the fifth lecture hall on Wednesday and Thursday. I hope you will take it on time.

Seeing the letter in the mailbox, Li Mo breathed a sigh of relief. After passing the exam, he can graduate directly and participate in the research project ahead of schedule.

Although Li Mo was very confident to 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 his head, only to see Ying Sasa standing behind her with a slight frown.

Yes, the Academic Affairs Office has approved my application, and the exam will be held this week. Li Mo said calmly.

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

Li Mo was slightly stunned, with a smile in the corner of his eyes, I'm already preparing to apply for Professor Wu's postgraduate entrance examination, and I will continue to stay at Yanda.

Really, that's great. Ying Sasha was overjoyed, perhaps realizing her gaffe, she said a little embarrassedly: Then you continue reading, I'll help them deliver the book.

Li Mo pondered for a long time as he looked at the back of Ying Sasa, who was walking lightly.

WeChat

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

Li Mo thought for a while, and replied: Thank you, please installers must work according to the drawings.

Zhou Dayong: Got it!

There are two bodyguards on site to supervise the work, maybe I can rest assured. Li Mo felt that he was a little weak, as the saying goes: You are covered in iron, how many nails can you drive?.

Wednesday morning, Classroom 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'm Professor Wu's graduate student, Zhou Ming. The boy with glasses greeted him shyly.

Li Mo smiled calmly, and said, It's a coincidence. I'm about to apply for Professor Wu's postgraduate entrance examination. Maybe you will be my senior brother in the future. Please take care of me.

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

I am over-flattered.

Cough, cough... The teacher from the Academic Affairs Office next to him saw that the two of them were talking about their homework, and interrupted, Because time is tight, let's start the exam now, student Li Mo.

This morning is expected to take the exam of 3 subjects, Mathematical Analysis, Advanced Algebra and Calculus Equation. Since the exam is in advance, it will not be carried out according to the normal exam time.

Before 12:00 noon, you just hand over the three completed test papers to me. He said and sent out the three test papers.

The first test paper is Mathematical Analysis\

,"1. Leaf-shaped line x=2t-t, y=2t-t, 0≤t≤2, find the area of ​​the graph surrounded 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, t 0 0.5 1 1.5 2 x 0 0.75 1 0.75 0 y 0 0.375 1 1.125 0, area A=∫\u003c0, 1\u003e(2t-t^41022)(2-2t)dt =∫\u003c0 , 1\u003e(4t-6t^2+2t^3)dt =(2t^2-2t^3+t^4/2)|\u003c0, 1\u003e=1/2.

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

This question is 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) at (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=(ydx+xdy)[cos(xy)]/[sin(xy)]

It took only 30 minutes for Li Mo to complete the Mathematical Analysis test paper. If it wasn't for the last open question, he used 6 methods to explain it, and it could be a little faster.

The next test paper is Advanced Algebra.

1. Let V1 and V2 be the solution spaces of the homogeneous equation system x1+x2++xn=0 and x1=x2==xn respectively, find V1, V2 and prove P^n=V1+V2, where P^n is a number An n-dimensional vector space over the field p.

Answer: V1 is the orthogonal complement space of the vector (1, 1, ., 1), the basis is (1, -1, 0, 0, . .,0),. . . , (1, 0,..., -1), the first component of each vector is 1, the k+1th component is -1, and the rest 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 two subspaces The sum is a straight sum, which happens to be the full space.

In 1 minute, the first question is completed. Since his wisdom has risen to level 2, he feels that he can easily grasp the problem-solving ideas.

Zhou Ming who was next to him saw that Li Mo had completed the Mathematical Analysis test paper, so he couldn't help but walked behind him and read it. I saw the immature boy in front of me, starting the topic like writing an article, chalking extremely fast.

Even if encountering tricky problems, the young man frowned slightly, and with a little thought, he could solve them easily.

Professor Wu often boasted in front of himself that there was a genius in the Department of Mathematics, but Zhou Ming didn't believe it. Who can enter the Department of Mathematics of Yanda University is not a genius.

But now seeing the young man who was solving the problem quickly, Zhou Ming really understood the meaning of genius.

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

The next test paper is Calculus Equation. Calculus Equation is famous for its large amount of calculation. It is not the kind of problem that can be easily solved with the idea of ​​solving the problem.

How long will it take to see you this time? Zhou Ming took a look at his mobile phone this time. It is now 9:30.

The first question, assuming that a, b, and c are all normal numbers, and y(x) is a solution of the differential equation ay''+by'+cy=0, verify:

lim(n~+∞)y(x)=0.

Li Mo quickly calculated in his mind and wrote:

ar^2+br+c=0

Because a, b, c\u003e0

The two roots are r1, r2

by Weida theorem

r1+r2=-b/a\u003c0

r1*r2=c/a\u003e0

If r1 and r2 are real roots, then obviously only r1 and r2\u003c0 can satisfy the sum less than zero and the product greater than zero

When x-\u003epositive infinity, exp(r1*x), exp(r2*x)-\u003e0, so y-\u003e0

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

So r1=m+in, r2=m-in

r1+r2=2m=-b/a\u003c0

because m\u003c0

exp(mx)-\u003e0 when x-\u003epositive infinity

|C1 cos n x+C2 sin nx|\u003c=|C1|+|C2|bounded

So when x-\u003epositive infinity y-\u003e0

In summary lim(n-\u003e+∞)y(x)=0

It's so long, and my hands are so tired. The calculus equations are indeed famous for being cumbersome. Li Mo cheered up.

Road 2. Road 3. Road 4.

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

Zhou Ming looked at his phone again. At 9:55, it took 25 minutes. The young man in front of him completed this set of test papers that he would have to complete in at least 2 hours.

Teacher, I'm done, can I hand in the paper in advance? Li Mo raised his hand and asked.

Finished? The teacher of the Academic Affairs Office who was in a daze at the door was a little surprised. There were 3 test papers. It was originally scheduled to be finished in one morning, and it already meant to test him.

Didn't expect him to finish it so quickly.

The teacher from the Academic Affairs Office came over, sorted out his test papers and put them in sealed bags.