It's notation:

means the first partial derivative of

with respect to

, that is,

while,

, the second partial derivative with respect to

.

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

, they had to show it happens for

.

Hence, the book found

by differentiating the

equation with respect to

twice, and then

in a similar way.

they then summed these two equations and found that it was zero. which means

is harmonic.

capice?