Find all harmonic polynomials u(x,y) of degree three, that is, u(x,y) = ax^3 + bx^2y + cxy^2 + dy^3

Question

Find all harmonic polynomials u(x,y) of degree three, that is, u(x,y) =  ax^3 + bx^2y + cxy^2 + dy^3

dumbcow · Answer

sorry im not familiar with these, I think the Laplacian needs to be zero which has to do the gradient but i havn't worked with these

im guessing this is diff equ or advanced calc 3

dumbcow · Answer

@ganeshie8 , @jim_thompson5910 , @SolomonZelman 

maybe they know more about harmonic polynomial

solomonzelman · Answer

nope, I am unfamiliar with harmonic polynomials, if I had a little more time, I would google it, but... sorry-:(

amistre64 · Answer

how is a harmonic poly defined? or is that what is given already?

amistre64 · Answer

the google on this doesnt ring any bells for me

dumbcow · Answer

http://en.wikipedia.org/wiki/Harmonic_polynomial http://en.wikipedia.org/wiki/Laplacian#Two_dimensions ok from this i think we got something take 2nd partial derivatives for x and y , then it equal to 0

curry · Answer

that's all the question gives.

dumbcow · Answer

$$\frac{du^2}{dx} = 6ax +2by$$
$$\frac{du^2}{dy} = 6dy +2cx$$
$$\Delta f = 6dy+2cx + 6ax +2by = 0$$
$$(c+3a)x + (b+3d)y = 0$$

curry · Answer

And the chapter doesn't give much about harmonic stuff.

curry · Answer

ye, i have that much so far too. But is that what defines all harmonic polynomials?

dumbcow · Answer

i dont know im just trying to figure it out like you haha

dumbcow · Answer

i found this link which may be helpful http://math.stackexchange.com/questions/100955/finding-general-harmonic-polynomial-of-form-ax3bx2ycxy2dy3?rq=1

curry · Answer

yes, i ran into that one as well

dumbcow · Answer

so we can conclude
c = -3a
b = -3d

sub that into original and theres your answer for general form

curry · Answer

so would that be the answer for all harmonic polynomials also?

dumbcow · Answer

well all of degree 3

curry · Answer

hmm, so we need to find it for degree 2,1, and 0 also?