國立彰化師範大學九十四學年度碩士班招生考試試題 系所:物理學系 科目: 物理數學 ☆☆請在答案紙上作答☆☆ 共 1 頁,第 1 頁 1. Evaluate the gradient , where is the scalar field x 2 y 2 2z 2 , working in both Cartesian and spherical polar coordinates and showing that they are equal. (16%) 2. The steady state temperature inside a bar satisfies the differential equation d dT ( (T ) ) 0 . dx dx The ends x 0 and x L are kept at the temperatures T 0 and T T0 0 , respectively. The thermal conductivity depends on the temperature according to (T) 0 T , where 0 and are two constants. (a) When integrating the steady state differential equation you will end up with a quadratic equation for T. Find this equation. (b) Solve this equation to find the steady state temperature distribution T ( x ) , 0 x L . (c) Which of the two roots is correct when 0 ? (18%) 3. A function u ( x, y) of two independent variables x and y satisfies the first order partial differential equation u ( x, y) u ( x, y) y u ( x , y) . x y By first looking for a separable solution of the form u ( x, y) X( x ) Y( y) , find the general x solution of the equation. u x x 3 when y x. 4. Evaluate the integral 0 Determine the u ( x, y) which satisfies the boundary condition (20%) cos t 2 dt . (10%) 0 1 2 5. Find the eigenvalues of the matrix A . Show that A I 2A ( I 1 2 is the corresponding unit matrix), and hence evaluate A 4 and A 8 . If t n is defined in terms of the trace of a matrix through t n [tr (A n )]1/ n , calculate t 2 , t 4 , and t 8 . Show that t n 2 1 as n . 6. Evaluate the Fourier transform e ax f (x) 0 1 2 g() x0 , and hence calculate x0 f ( x )e ix dx of the function (a 0) g() d . -1- (20%) 2 (16%)