Zorich mappig:

Izračunajmo Jakobijan od Zorich mapping $$ J= \left | \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3} \end{array} \right | = e^{3x_3}\frac{x_1\sin x_1}{x_1 ^2+x_2 ^2} $$ $$\left | \left | \left( f_1 , f_2 , f_3 \right ) \right | \right | ^2 = e^{2x_3} $$ Izračunajmo kvadrate normi parcijalnih izvoda \begin{eqnarray} & & \left | \left | \left( \frac{\partial f_1}{\partial x_1} , \frac{\partial f_2}{\partial x_1} , \frac{\partial f_3}{\partial x_1}\right ) \right | \right | ^2 = e^{2x_3} \left( 1+\frac{x_2 ^2 \sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2}\right ) \nonumber \\ & & \left | \left | \left( \frac{\partial f_1}{\partial x_2} , \frac{\partial f_2}{\partial x_2} , \frac{\partial f_3}{\partial x_2}\right ) \right | \right | ^2 = e^{2x_3} \frac{x_1 ^2 \sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} \nonumber \\ & & \left | \left | \left( \frac{\partial f_1}{\partial x_3} , \frac{\partial f_2}{\partial x_3} , \frac{\partial f_3}{\partial x_3}\right ) \right | \right | ^2 = e^{2x_3} \nonumber \end{eqnarray} Izračunajmo proizvode \begin{eqnarray} & & \left < \left( f_1 , f_2 , f_3 \right ), \left( \frac{\partial f_1}{\partial x_1} , \frac{\partial f_2}{\partial x_1} , \frac{\partial f_3}{\partial x_1}\right ) \right >= 0 \nonumber \\ & & \left < \left( f_1 , f_2 , f_3 \right ), \left( \frac{\partial f_1}{\partial x_2} , \frac{\partial f_2}{\partial x_2} , \frac{\partial f_3}{\partial x_2}\right ) \right >= 0 \nonumber \\ & & \left < \left( f_1 , f_2 , f_3 \right ), \left( \frac{\partial f_1}{\partial x_3} , \frac{\partial f_2}{\partial x_3} , \frac{\partial f_3}{\partial x_3}\right ) \right >= e^{2x_3} \nonumber \end{eqnarray} Preslikavanje $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ je $k-$guasiregular ako postoji konstanta $k$ takva da je $$||D_f (x_1,x_2,x_3)||^3 \leq k |J_f (x_1, x_2, x_3)| .$$ Definicija norme matrice $A$ $$||A||=\sup\Big\{||A(h)|| \Big | ||h||=1\Big\}$$ $$ Z' (x_1, x_2, x_3)= \left [ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3} \end{array} \right ] = e^{3x_3}\frac{x_1\sin x_1}{x_1 ^2+x_2 ^2} $$ $$ Z' (x_1, x_2, x_3) \left ( \begin{array}{ccc} h_1 \\ h_2 \\ h_3 \end{array} \right ) = \left [ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3} \end{array} \right ] \left [ \begin{array}{ccc} h_1 \\ h_2 \\ h_3 \end{array} \right ] $$ $$ Z' (h_1, h_2, h_3)= \left [ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1}h_1 & \frac{\partial f_1}{\partial x_2}h_2 & \frac{\partial f_1}{\partial x_3}h_3 \\ \frac{\partial f_2}{\partial x_1}h_1 & \frac{\partial f_2}{\partial x_2}h_2 & \frac{\partial f_2}{\partial x_3}h_3 \\ \frac{\partial f_3}{\partial x_1}h_1 & \frac{\partial f_3}{\partial x_2}h_2 & \frac{\partial f_3}{\partial x_3} h_3 \end{array} \right ] $$ $$\left| \left| Z' (x_1, x_2, x_3) (h_1, h_2, h_3) \right| \right|^2 = e^{2x_3} \left( h_1 ^2 \left( 1+\frac{x_2 ^2 \sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} \right) + h_2 ^2 \frac{x_1 ^2 \sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} +h_3 ^2 -2h_1 h_2 \frac{x_1 x_2 \sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} \right)$$ $$||Z'||^2=\sup\Big \{||Z'(h_1, h_2, h_3)||^2 \Big | h_1 ^2 + h_2 ^2 + h_3 ^2 =1 \Big \}$$ $$h_3 ^2=1- h_1 ^2 - h_2 ^2$$ \begin{eqnarray} & \left| \left| Z' (h_1, h_2, h_3) \right| \right|^2 & = e^{2x_3} \left( \frac{\sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} \left(h_1x_2-h_2x_1\right)^2 - h^2_2 +1 \right) \nonumber \\ & & \leq e^{2x_3} \left( \frac{\sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} h_1^2 x_2^2 +1 \right) \nonumber \\ & & \leq e^{2x_3} \left( \frac{x_2 ^2\sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} +1 \right) , (h_1 =1, h_2=0, h_3=0) \nonumber \end{eqnarray} Pa je uslov kvaziregularnosti \begin{eqnarray} ||Z' (x_1,x_2,x_3)||^3 & \leq & k |J(x_1, x_2, x_3)| \nonumber \\ \left( e^{2x_3} \left( \frac{x_2 ^2\sin ^2 x_1}{(x_1 ^2 + x_2 ^2)^2} +1 \right) \right)^ {\frac{3}{2}} & \leq & k e^{3x_3} \left( \frac{x_1 \sin x_1}{x_1 ^2 + x_2 ^2} \right) \nonumber \end{eqnarray}