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 : 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 --

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