Originally Posted by

**Jhevon** It's notation: $\displaystyle \displaystyle v_x$ means the first partial derivative of $\displaystyle \displaystyle v$ with respect to $\displaystyle \displaystyle x$, that is, $\displaystyle \displaystyle \frac {\partial v}{\partial x}$

while, $\displaystyle \displaystyle v_{xx} = \frac {\partial ^2 v}{\partial x^2}$, the second partial derivative with respect to $\displaystyle \displaystyle x$.

you can figure out the rest.

I'm afraid your work is incorrect and completely misses the point. In fact, you almost go backwards....or maybe you're just mislabeling u and v...? at the very least, what you did was completely confusing. what are you differentiating with respect to?? functions of x and y, rather than just x and y themselves??

Harmonic means, as they say, the second order un-mixed partials sum to zero. so, assuming this happens for $\displaystyle \displaystyle v$, they had to show it happens for $\displaystyle \displaystyle u$.

Hence, the book found $\displaystyle \displaystyle u_{xx}$ by differentiating the $\displaystyle \displaystyle u(x,y)$ equation with respect to $\displaystyle \displaystyle x$ twice, and then $\displaystyle \displaystyle u_{yy}$ in a similar way.

they then summed these two equations and found that it was zero. which means $\displaystyle \displaystyle u(x,y)$ is harmonic.

capice?