Funktioiden $$$\left\{x = 6 u + v, y = 9 u - v\right\}$$$ Jacobin matriisi ja sen determinantti

Laske $$$\left\{x = 6 u + v, y = 9 u - v\right\}$$$:n Jakobiaani.

Jakobin matriisi määritellään seuraavasti: $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right].$$$

Meidän tapauksessamme pätee $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial}{\partial u} \left(6 u + v\right) & \frac{\partial}{\partial v} \left(6 u + v\right)\\\frac{\partial}{\partial u} \left(9 u - v\right) & \frac{\partial}{\partial v} \left(9 u - v\right)\end{array}\right].$$$

Laske derivaatat (vaiheet: katso derivointilaskuri): $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}6 & 1\\9 & -1\end{array}\right]$$$.

Jacobin determinantti on Jacobin matriisin determinantti: $$$\left|\begin{array}{cc}6 & 1\\9 & -1\end{array}\right| = -15$$$ (vaiheet, ks. determinanttilaskin).

Jacobin matriisi on $$$\left[\begin{array}{cc}6 & 1\\9 & -1\end{array}\right]$$$A.

Jacobin determinantti on $$$-15$$$A.