If $u = x{y^2}{\tan ^{ - 1}}\left( {\frac{y}{x}} \right)$,then $x{u_x} + y{u_y} = $

If $z = y + f(v)$,where $v = \frac{x}{y}$,then $v \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}$ is

If $u = (x^2 + y^2 + z^2)^{3/2}$,then $\left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial u}{\partial y} \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 = $

If $u = u(x, y) = \sin(y + ax) - (y + ax)^2$, then which of the following is true?

If $u(x,y) = y \log x + x \log y$,then ${u_x}{u_y} - {u_x} \log x - {u_y} \log y + \log x \log y = $

$\begin{aligned} & f(x, y)=2(x-y)^2-x^4-y^4 \\ & \left|\left(f_{x x} f_{y y}-f_{x y}^2\right)\right|_{(0,0)} \end{aligned}$