PHYS 325
HW#1
F08
1. Write out aij x j xk in expanded form, assuming aik  aki , and i,k=1,2,3.
2. Find the values of  ij ij ,  ij jk km im ,  ik ikm ,  jkl Ak Al all indices ranging from 1
to 3.
3. Evaluate  ikl jkl and ijk ijk all indices ranging from 1 to 3.
4. Use the suffix notation to show that
 
 


a) div AxB  B.curlA  A.curlB
 
       
b) curl AXB  AdivB  BdivA  B. A  A. B
5. Using the suffix notation
a) Consider the scalar function

   
hr   r  A . r xB



where A and B are constant vectors. Show that the gradient of hr  is
     

h  Ax r xB  Bx r xA .

 1
A.r
b) Show that A.    3 .
r
r
6. Calculate, using the suffix notation,




p.r
a)  , where  r   3 and p is a constant vector.
r
 
 m
 
xr

b) xA , where A  3 and m is a constant vector. r  xi xi
r
7. Verify that if   xi  satisfies the equation 2  K 2  0 where K is a constant then
the second-order tensor
Tik  (i k   ik  2 )
is a solution of
 ijk ilm jmTkp  K 2Tlp  0





   
 
 
 


  



8. Let 1 = i + j,  2 = i − j,  3 = j + k. Find  1 ,  2 ,  3 , (gij) and
gij. If v = i+2j + k, find vi and vj, the contravariant and covariant components of the
 
 
vector. Is this an orthogonal coordinate system? ( Hint: gij=  i .  j , gij=  i . j and
gik gkj = δij )
9.
a)
b)
c)
Show that,
AijBiCj is a scalar,
AijkBij is a contravariant tensor of rank 1 (contravariant vector),
AijkBi is a covariant tensor of rank 2.