How do I verify that v is a harmonic function?

Question

My question is "verify that v defined by

v = 49/40 - 1/5(x^4 - 6x^2y^2 + y^4) is harmonic and that, when |x|<1 and |y|=1, -0.025 <= v - 1 - x^2 <= 0.025".

How do I show the function is harmonic? I think I have to transform it into a sine/cosine function but I don't know how.

Answer 1

calculate d^2x/dx^2 and d^2y/dy^2 ( partial derivatives)
and sum.The sum should give 0
v=49/40 -1/5(x^4-6x^2+1)
Let´s find the the extremes of z=49/40-1/5(x^4-6x^2+1)-1-x^2
z´= -1/5(4x^3-12x)-2x =0
x(-4/5x^2+2/5)= 0
x=0 and x= +-sqrt(2/2)x=0 local minimum
z(0)=0.025
z(1/aqrt2) =0.075
0.025 <= v-1-x^2<= 0.075. (check calculus)

Answer 2

A harmoic function just means that it is twice differentiable and you can use partials to differenciate this twice. That should work. If you can not do it email me and I will work it out but because they are squared values it will work out for you.

Answer 3

You can like play the harmonica? How hard do you have to blow?