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stuck on show that u= x/(x^2+y^2) is harmonic?
show that u= x/(x^2+y^2) is harmonic
i know you have to take the partial derivative of x an y, and some how there supposed to equal 0 when you add them together to prove it is harmonic…but i don't think im doing it right
1 Answer
- kbLv 710 years agoFavorite Answer
u_x = [1(x^2 + y^2) - x * 2x]/(x^2 + y^2)^2 = (y^2 - x^2) / (x^2 + y^2)^2
u_xx = [(-2x)(x^2 + y^2)^2 - (y^2 - x^2) * 2(x^2 + y^2)2x]/(x^2 + y^2)^4
........= -2x (3y^2 - x^2) / (x^2 + y^2)^3
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u_y = -2xy/(x^2 + y^2)^2
u_yy = [(-2x)(x^2 + y^2)^2 - (-2xy) * 2(x^2 + y^2) * 2y] / (x^2 + y^2)^4
........= 2x (-x^2 + 3y^2) / (x^2 + y^2)^3
Therefore, u_xx + u_yy = 0, as required.
Hence, u is harmonic.
I hope this helps!
