課程名稱︰化學數學一
課程性質︰必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2007/01/16
考試時限（分鐘）：100 min
是否需發放獎勵金：是
試題 :
1.(20 pts) Consider the 2D Laplacian ▽^2=(對x 二次偏微+對y二次偏微).
Derive the formula for ▽^2 in terms of the polar coordinate (r,θ),
where r= (x^2 + y^2)^1/2 , and tan θ= y/x
2. (40 pts)
(a) Find the general golution of the PDE
δP δ^2
i __ Ψ(x,t) = - ____ Ψ(x,t)
δt δx^2
for -π<= x <= π with the boundary condtions
Ψ(-π,t)=Ψ(π,t) and
δ δ
__ Ψ(-π,t) = ____ Ψ(π,t)
δx δx
(b) Find the particular solution which further satisfies the intial
condition Ψ(x,0)= cosx * sin(4x + π/3)
3.(30 pts)
(a) Find the general solution of the ODEs x''(t)=-5x(t)+√3 y(t)
and y''=√3x(t) -7 y(t)
(b) Find the particular solution which further satisfies
x(0)=1, x'(0)= 0, y(0)=0 and y'(0)=1
4.(15 pts) Let A(a matrix) be a real symmetric matrix. Let u and v be the
eigenvectors of A and their eigenvalues are not the same. Show that u and v
are orthorgonal.
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