104年 - 104 國立中山大學_碩士班招生考試_電機系(甲組、丁組、己組)：工程數學甲#110231

(a) (10%) Define the thermal energyu(t,x)da. Show that under the above assumptions, T(t)

is constant in time; i.e., f(x)dr, for all t≥0.

(b) (10%) Let f(x) = cos(πx). Find the solution u.

(b). (15%) For the values found in (a), ind a surface S with the following property: the path integral F ㆍ dr is equal to O for any two points P, Q (connected by any curve C) on the surface S.

(a) Suppose ItaA is nonsingular, thus, for any nonzero scalar a, Ωa := is a well-definedmatrix. Then we know from knowledge of eigensystem of a square matrix that, corresponding to any. What is the mathematical relation betweenλ and μ?(3%)

(b) If Ω₁ , that is , is an orthogonal matrix, then what mathematical relation between A and A^Tcan be derived? (5%)

(c) If Ω_a. is an idempotent matrix, then what are all possible values of det A ? (5%)

(a) Write out the set,where N(●)and R(●) indicate the null space and the range of a matrix, respectively. (5%)

(b) Consider the inner product space V =, where ＜A,B) :=tr(^ATB) for A and B in . Describe Sh as the span of a set of orthonormal vectors in  . (7%)

(a)(5%) Compute the following integral
, n is a positive integer

(b)(10%) Suppose the answer you obtained in Part (a) is jnt . Use Part (a) to evaluate

Problem 6(15%) Define the Fourier transform of a signal dt, and its inverse Fourier ransform is . It is already known that the Fourier transform of signal x(i)=sin(ai) /(πi) is and 9.Compute the Fourier transform of the signal