Prove that $\arctan({y \over x})$ does not conform to $yZ_x=xZ_y$ in the first quarter namely $D=\{(x,y)|x>0,y>0 \}$.
I guess this can be proved straightforwardly from the calculation: $$ Z_x={dz \over dx}=-\frac{y}{x^2+y^2}\\ Z_y={dz \over dy}=\frac{x}{x^2+y^2} $$ Therefore: $$ yZ_x=xZ_y \Leftrightarrow y\cdot -\frac{y}{x^2+y^2}=x\cdot \frac{x}{x^2+y^2} $$
Which is a contradiction. What puzzles me why does the exercise say "in the first quarter"? As far as I see this inequality will hold in any quarter.