\textbf{1.} Ov��te rovnost
    $$
        \frac{\partial^2 u}{\partial x \partial y} =
        \frac{\partial^2 u}{\partial y \partial x}
    $$ pro funkci $u=x^2 - 2xy -3y^2.$\\
    \textbf{�e�en�.} Spo�teme $\frac{\partial^2 u}{\partial x \partial y}:$
    $$
        \frac{\partial^2 u}{\partial x \partial y} =
        \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial y}\right)
        = \frac{\partial}{\partial x} (-2x -6y) = -2.
    $$
    D�le spo�teme $\frac{\partial^2 u}{\partial y \partial x}:$
    $$
        \frac{\partial^2 u}{\partial y \partial x} =
        \frac{\partial}{\partial y} \left(\frac{\partial u}{\partial x}\right)
        = \frac{\partial}{\partial y} (2x-2y) = -2.
    $$
    Vid�me, �e rovnost skute�n� plat�.
