Solutions to Tensor Problems

1)If  1  =  i^ i^ + j^ j^ + k^ k^

Write out the expression for 1 × D   


Ans.

1 × D  =  i^ D_x + j^ D_y + k^ D_z  

2. a) Provide a matrix which satisfies:  i^ i^ + j^ j^ + k^ k^  = 7/2 

The trace must equal to 7/2 so the diagonal elements need to yield that sum. Thus:

(1/2......0........0)

(0........1........0)

(0........0....... 2)

is one example of a matrix that satisfies the condition

b) Write out the trace for the metric tensor.

Ans. The metric tensor is:  

g _ik      =

(1.....0...............0)

(0.....r²...............0)

(0.....0........r² sin f)

Then the trace (Tr) =

1 +   r²  +     r² (sin f)=   1 + r²  (1 + sin f) 

c) Give one example of  3 x 3  tensor, then show how it might contain an anti-symmetric and symmetric  tensor (also how to go from one form to the other). 

Ans.  Let  a _{i j}   =

(2......3........2)

(5........7.......-2)

(4........-4....... 0)

Then the antisymmetric form is:   a _{j  i}  =

(0......3........-4)

(-3.......0.......-2)

(4........2....... 0)

Then the symmetric part  is the difference:  a _{i j}  - a _{j  i}  =

(2......0........6)

(8......7........0)

(0......-6....... 0)

3. Find the trace of each matrix given:

Soln. simply add the diagonal elements for each, i.e.

a)  Tr = 2 + 8 p  +  p  =   2 + 9 p

b) Tr =    2 r +  2 L  +   r /2  =  5 r /2  + 2L  

c) Tr = p +  p/ 3   + p/ 6   + p/7 =   69p / 42