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How do I verify that the function $$u(x,y)=x^2-y^2-y$$ is harmonic

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What is the definition of harmonic? Do you know how to take partial derivatives? – wj32 Nov 10 '12 at 11:57
What's the definition of a function being harmonic? Check that your $u$ satisfies it. – mrf Nov 10 '12 at 11:57

$$u_{xx}=2\;\;,\;u_{yy}=-2\Longrightarrow \Delta u=...?$$

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It is the real part of $z^2 +iz$, therefore harmonic.

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Let $u(x,y)=ax^2+2hxy+by^2+2gx+2fy+c=0$

So, $u_x=2ax+2hy+2g, u_{xx}=2a$

Similarly, $u_{yy}=2b$

Using this, we need $u_{xx}+u_{yy}=0$ and $u_{xx}+u_{yy}=2(a+b)\implies b=-a$

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