Proving the Order of a Tensor: Transformation Laws and Quotient Theorem

[rarkup]

Homework Statement

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

Homework Equations

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.

 

[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 a_ij? This will tell you how a_ij are related to a'_ij, i.e. how the components of a transform.

 

[rarkup]

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 a_ij? This will tell you how a_ij are related to a'_ij, i.e. how the components of a transform.

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.

 

[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.