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題組內容

7.

(a) Apply the divergence theorem to evaluate and S is the surface of the region bounded by the cylinder: r≤5, 0≤ θ ≤ 2π, 0≤ Z≤ 4. (5%)

其他申論題

(a) Find the determinant of M and obtain the inverse matrix .(3%+4%)

(b) Estimate the eigenvalues and eigenvectors of M.(4%+4%)

(a) Let f(x,y) = asin(xy) +x2 +4y2(1 -y). For what values of a will f have a local minimum at (0,0)? (5%)

(b) Let (a≥0). Find I'(a). It is evident that I(0) = 0. Solve I(a). (5%)

(b) Let F =Fㆍdr where C is a counterclockwise circle x2 + y2 = 9 on the xy plane. (5%) [Formula] Divergence and curl in cylindrical coordinates:

(a)Let A0 ＞0. The Fourier integral representation of b(ω) sin wx]dx. Find a(ω) and b(ω). (5%)

(b) Consider an infinite beam problem Elu"" + ku = f(x) with the loading f(x) given in (a). The deflection u can be solved and written as u(x)= d(ω) sin wxJdx. Find c(ω) and d(ω). (5%)

9. Solve the Dirichlet problem in polar coordinates:,u(1,θ)=0,u(2,θ)=10. (10%)

(a) Evaluatedz where C is the counterclockwise circle |z| = 4. (5%)

(b) Evaluate the integral by residue theorem. (5%)

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