推 a27518751 :超難的喇 11/11 01:53
課程名稱︰化學數學一
課程性質︰化學系必帶
課程教師︰陸駿逸副教授
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2010/11/10
考試時限（分鐘）：140分鐘
是否需發放獎勵金：是
試題 :
http://0rz.tw/WDpvY
Chemical Mathematics Midterm Examination
2010/11/10 10:20~12:10(後來延長到12:40)
π
1(5pts)Calculate the imaginary part of the complex number i
2(20pts)Given M=┌ ┐
│ 1 2i│
│-2i 1 │
└ ┘
Construst the unitary transform U which diagonalizes M by the similar
-1 30
transformation U MU = D Calculate U D and M
3(20pts)Three masses m1=1 m2=2 m3=3 are connected by two springs along the
x axis,with the spring constants k1=10 k2=15
Consider the motion along the x axis. What are the vibrational frequencies?
What are the ratio of the atomic displacements?
4(20pts)Use the Gram-Schmidt method to find an orthogonal basis of the vector
space〈real coefficient polynomials with the power not greater than three〉.
Here the inner product is defined as：
2
∞ -x
〈f│g〉=∫ e f(x)g(x)dx
-∞
Hint: you can make use of the following integrals
2 2 2
∞ -x __ ∞ -x 2 __ ∞ -x 4 __
∫ e dx =√π ∫ e x dx =(√π)/2 ∫ e x dx =(3√π)/4
-∞ -∞ -∞
5(20pts)Find the first three Fourier coefficients(a0,a1,b1)of the function f.
f(x)=a0/2 + a1cosx + a2cos2x +.....+ b1sinx + b2sin2x +.......
where f(x)=x for x>0 and f(x)=0 for x<0
_______
6(10pts)What is the gradient of F(x,y)=r, where r=√x^2+y^2
2 2 2 2 2 2
7(20pts)Given the F(x,y,z)=3x + y- 4yz + 4z, and the constraint x + y + z = 2
Find the extrema of F
ans http://0rz.tw/opU8n
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