What Are All Second-Degree Harmonic Polynomials in R2?

[Somefantastik]

In R², I am to find all homog. polys (deg 2) that are harmonic.

the earlier homework included something like u = xy, show it's harmonic. EASY as pi. But I'm not really sure how to set this problem up. I understand the concept that a harmonic function will look like $$\nabla^{2} u = 0$$, but I'm not sure how to find all polys of degree 2.

I started out doing something like

$$u = a_{2}x^{2}_{1} + a_{1}x_{1} + a_{0} + b_{2}x^{2}_{2} + b_{1}x_{2} + b_{0}+...$$

and taking the partial with respect to each x_i but that's not getting me very far.

Any suggestions?

 

[Ben Niehoff]

If you are in R^2, then you only need x and y. The general 2nd-degree polynomial is simply

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F$$

 

[Somefantastik]

Thank you. I ended up with f(x,y) = Ax² + Bx + C - Ay² + Ey + F covers harmonic polys in R².