Proof of transformational symmetry

[Hendrick]

Homework Statement

Using indical notation, prove that D retains it's symmetry when transformed into any other coordinate system, i.e. $$D'_{pq} = D'_{qp}$$ (where D is a symmetric 2nd order tensor)

Homework Equations

$$D'_{pq} = a_{pr}a_{qs}D_{rs}$$ (law of transformation for 2nd order tensors)

The Attempt at a Solution

$$D_{pq} = D_{qp}$$ (as D is symmetric)

$$D'_{pq} = a_{pr}a_{qs}D_{rs}$$
$$D'^{T}_{pq}=(a_{pr}a_{qs}D_{rs})^{T}$$
$$D'_{qp} = a_{qs}a_{pr}D_{sr}$$ (can someone please explain why when you transpose this, the a's swaps position but the D swaps indices?)
$$D'_{pq} =a_{pr}a_{qs}D_{rs}$$ (swapping p<=>q, s<=>r)

We can see that $$D'_{pq}$$ is in the same form of $$D'_{qp}$$, thus $$D'_{pq} = D'_{qp}$$

 

[Hendrick]

Anyone?