In the two-dimensional case, show that the homogeneous cubic form i,j,kwiwjwk can be written, using power

In the two-dimensional case, show that the homogeneous cubic form κ

i,j,kwiwjwk can be written, using power notation, in the form Q3 (w) = κ30w3 1 + κ03w3 2 + 3κ21w2 1w2 + 3κ12w1w2 2

.

By transforming to polar coordinates, show that

κr = − (r − 1)!∑

i a

r i = − (r − 1)! < r > (r < n)

Xj+1 = {

Yj+1 = {

Xj − Zj if Xj > Zj

ϵj+1 otherwise Yj − Zj if Yj > Zj

ϵ

′

j+1 otherwise Q3 (w) = r 3 {τ1 cos(θ − ϵ1) + τ3 cos(3θ − 3ϵ3)}, where Find similar expressions for ϵ1 and ϵ3 in terms of the κs.