Prove order of tensor

1. Sep 13, 2009

rarkup

1. The problem statement, all variables and given/known data

The equation of a quadric surface takes the form

$$a_i_jx_ix_j = 1, a_i_j = a_j_i$$

relative to the standard coordinate axes. Under a rotation of axes the equation of the surface becomes

$$a'_i_jx'_ix'_j = 1$$

By considering the coordinates $$x_i$$ as components of position vectors, show that the coefficients $$a_i_j$$ are components of a second order tensor

(i) using transformation laws;
(ii) using the quotient theorem

2. Relevant equations

3. The attempt at a solution

I need a bit of a kick-start on this one.
I'm guessing my objective will be to show that:

$$a'_i_j = l_i_ml_j_na_m_n$$ where $$l_i_j$$ is an entry in the matrix representing the rotation.

but it's not clear to me how to relate a, l and x.

2. Sep 14, 2009

dx

In the equation a'ijx'ix'j = 1, write x' in terms of x. In the resulting equation, can you identify an expressionis that must be equal to aij? This will tell you how aij are related to a'ij, i.e. how the components of a transform.

3. Sep 14, 2009

rarkup

Thanks for your response. Rewriting x' in terms of x I get:

$$a_i_j'x_i'x_j' = 1 => a_i_j'l_i_mx_ml_j_nx_n = 1 = a_m_nx_mx_n$$
$$=> a_i_j'l_i_ml_j_n = a_m_n$$

This is close, but it's not clear to me how to shuffle the elements of the rotation matrix L from the LHS to the RHS. I also suspect the symmetry of a was mentioned for a reason, and that it's likely that it suggests a property that I should be taking advantage of.

4. Sep 15, 2009

rarkup

Would really appreciate some help on this. I'm studying remotely and have no other students to refer to, and the course tutor seems to be on extended leave. I have about 24 hours before the last mail pick-up to make the assignment due date. Any help fully appreciated.