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.