國立成功大學113學年度碩士班招生考試試題
系所:土木工程學系
考試科目:工程數學
解答:(a)A=[83−32]⇒det(A−λI)=(8−λ)(2−λ)+9=λ2−10λ+25=0⇒(λ−5)2=0⇒λ=5⇒A has eigenvalue λ1=5 of multiplicity two.(b)c0+c1λ有兩個變數c0及c1,而λ1是重根,無法完整求出兩變數(c){f(A)=Anr(A)=c1A+c0I⇒{f(λ)=λnr(λ)=c1λ+c0⇒{f(λ)=r(λ)f′(λ)=nλn−1=r′(λ)=c1⇒{f(5)=r(5)f′(5)=r′(5)⇒{5n=5c1+c0n⋅5n−1=c1⇒c0=(1−n)5n⇒An=n⋅5n−1A+(1−n)5nI=n⋅5n−1[83−32]+(1−n)5n[1001]=[8n5n−1+(1−n)5n3n5n−1−3n5n−12n5n−1+(1−n)5n]⇒An=[(3n+5)5n−13n5n−1−3n5n−1(5−3n)5n−1](d)n=9⇒A9=[32⋅5827⋅58−27⋅58−22⋅58]
解答:(a)gv=yzex2+y2+z2i+xzex2+y2+z2j+xyex2+y2+z2k⇒div(gv)=∂∂xyzex2+y2+z2+∂∂yxzex2+y2+z2+∂∂zxyex2+y2+z2=2xyzex2+y2+z2+2xyzex2+y2+z2+2xyzex2+y2+z2=6xyzex2+y2+z2(b)∇2g=∂2∂x2g+∂2∂y2g+∂2∂z2g=∂∂x2xex2+y2+z2+∂∂y2yex2+y2+z2+∂∂z2zex2+y2+z2=2ex2+y2+z2+4x2ex2+y2+z2+2ex2+y2+z2+4y2ex2+y2+z2+2ex2+y2+z2+4z2ex2+y2+z2=(6+4(x2+y2+z2))ex2+y2+z2(c)grad g=(∂∂xg,∂∂yg,∂∂zg)=(2xex2+y2+z2,2yex2+y2+z2,2zex2+y2+z2)⇒curl(grad g)=|ijk∂∂x∂∂y∂∂z2xex2+y2+z22yex2+y2+z22zex2+y2+z2|=0(d)curl v=|ijk∂∂x∂∂y∂∂zyzxzxy|=0⇒div(curl v)=0(e)v⋅v=y2z2+x2z2+x2y2⇒grad(v⋅v)=2x(y2+z2)i+2y(x2+z2)j+2z(x2+y2)k
解答:A(α)=∫∞−∞f(x)cos(αx)dx=∫30cos(αx)dx=[1αsin(αx)]|30=1αsin(3α)B(α)=∫∞−∞f(x)sin(αx)dx=∫30sin(αx)dx=[−1αcos(αx)]|30=1α(1−cos(3α)⇒f(x)=1π∫∞0(A(α)cos(αx)+B(α)sin(αx))dα=1π∫∞0(1αsin(3α)cos(αx)+1α(1−cos(3α)sin(αx))dα=1π∫∞01α(1+sin(3α−αx))dα
解答:A=[33−62−2411−2]R2−(2/3)R1→R2→[33−60−4811−2]R3−(1/3)R1→R2→[33−60−48000]⇒A=[1002/3101/301][33−60−48000]⇒L=[1002/3101/301],U=[33−60−48000]
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解題僅供參考,其他歷年試題及詳解
小錯誤,糾正一下,
回覆刪除2(d):A^9中的a_(2,2)=-22*5^8
3(b):∇^2(g)=(6+4(x^2+y^2+z^2))*e^(x^2+y^2+z^2).
謝謝!
謝謝,已修訂
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