**1)If**

**= i**

__1__**^ 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.....r

(0.....0........r

(0.....r

^{2}...............0)(0.....0........r

^{2}sin**f**

**)**

**Then the trace (Tr) =**

**1 +**

**r**

^{2}**+**

**r**

^{2}(sin**f**

**)**

**= 1 +**

**r**

^{2}

**(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 +**

**2**

**L**

_{}**c) Tr =**

**p +**

**p/ 3 +**

**p/ 6 +**

**p/7 = 69**

**p / 42**

_{}
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