Jacobimatrix/sin x durch x^2+y^4, x^2y durch x^2+y^2, ln (x^2+y^2)/Aufgabe/Lösung

Die partiellen Ableitungen sind

\frac{\partial f_{1}}{\partial x} = \frac{(x^{2} + y^{4}) \cos x - 2 x \sin x}{(x^{2} + y^{4})^{2}},

\frac{\partial f_{1}}{\partial y} = \frac{- 4 y^{3} \sin x}{(x^{2} + y^{4})^{2}},

\frac{\partial f_{2}}{\partial x} = \frac{2 x y (x^{2} + y^{2}) - x^{2} y (2 x)}{(x^{2} + y^{2})^{2}} = \frac{2 x y^{3}}{(x^{2} + y^{2})^{2}},

\frac{\partial f_{2}}{\partial y} = \frac{x^{2} (x^{2} + y^{2}) - x^{2} y (2 y)}{(x^{2} + y^{2})^{2}} = \frac{x^{4} - x^{2} y^{2}}{(x^{2} + y^{2})^{2}},

\frac{\partial f_{3}}{\partial x} = \frac{2 x}{x^{2} + y^{2}}

und

\frac{\partial f_{3}}{\partial y} = \frac{2 y}{x^{2} + y^{2}} .

Somit ist die Jacobi-Matrix in einem Punkt $(x, y)$ gleich

(\begin{matrix} \frac{(x^{2} + y^{4}) \cos x - 2 x \sin x}{(x^{2} + y^{4})^{2}} & \frac{- 4 y^{3} \sin x}{(x^{2} + y^{4})^{2}} \\ \frac{2 x y^{3}}{(x^{2} + y^{2})^{2}} & \frac{x^{4} - x^{2} y^{2}}{(x^{2} + y^{2})^{2}} \\ \frac{2 x}{x^{2} + y^{2}} & \frac{2 y}{x^{2} + y^{2}} \end{matrix}) .