In the two-dimensional case, show that the homogeneous quartic form i,j,k,lwiwjwkwl can be written, using power

In the two-dimensional case, show that the homogeneous quartic form

κ

i,j,k,lwiwjwkwl can be written, using power notation, in the form Q4 (w) = κ40w4 1 + κ04w4 2 + 4κ31w 3

1w2 + 4κ13w1w 3

2 + 6κ22w2 1w2 2

.

By transforming to polar c∞rdinates, show that Q4 (w) = r 4 {τ0 + τ2 cos(2θ − 2ϵ2) + τ4 cos(4θ − 4ϵ4)}.

Show that 8τ0 = 3κ40 + 3κ04 + 6κ22 is invariant under orthogonal transformation of X.

Find similar expressions for τ2, τ4, ϵ2 and ϵ4 in terms of the κs.