
 
\subsubsection {}
Doka�te, �e funkce $z = f(x,y)$ je zad�na v n�jak�m okol� bodu 
$(1,-1,1)$ jednozna�n� rovnic� $z^2 x^4 + y^3 z + (x^2 + y^2) z^4 - 2 = 
0$ a spo�t�te v�echny prvn� a druh� parci�ln� derivace $f$ v $(1,-1)$.

(b)  Doka�te, �e funkce $f_1(x), f_2(x)$ jsou zad�ny rovnicemi 
$$ x^3 (y + z) + y^5 + z^4 - 4 = 0,$$
$$ x (y - z) + y^4 + z^5 - 2 = 0$$
jednozna�n� v okol� 
bodu $(1,1,1)$ a spo�t�te $f_1'(1), f_1''(1), f_2'(1), f_2''(1).$

(c)  Spo�t�te $y'(1), y''(1)$, kde $y$ je zad�na rovnic� $x^y = y^x$ 
okolo $(1,1)$.

\subsubsection {}
(a) Spo�t�te $\frac{\partial^2 z}{\partial x\partial y}$ "v $u = 
2, v = 1$", jestli�e $x = u+v^2, y = u^2 - v^3, z = 2uv.$

(b) P�eve�te do pol�rn�ch sou�adnic v�raz $x \frac{\partial u}{\partial 
y} + y \frac{\partial u}{\partial x}.$

(c) P�eve�te do pol�rn�ch sou�adnic v�raz $\frac{\partial^2 u}{\partial 
x^2} + \frac{\partial^2 u}{\partial y^2}.$

Pol�rn� sou�adnice jsou $r, \alpha$, kde $x = r \cos \alpha, y = r \sin 
\alpha, r > 0, \alpha \in {\Bbb R }.$ Hled�me vyj�d�en� pomoc� 
$\overline{u}(r,\alpha) = u(r \cos\alpha, r \sin \alpha).$
