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() 
x0
, and hence calculate
x0






f ( x )e ix dx of the function (a  0)
g() d .
-1-
(20%)
2
(16%)