Jacobian dan determinannya dari $$$\left\{u = x, v = y, w = x y\right\}$$$
Masukan Anda
Hitung Jakobian dari $$$\left\{u = x, v = y, w = x y\right\}$$$.
Solusi
Matriks Jacobian didefinisikan sebagai berikut: $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y}\end{array}\right].$$$
Dalam kasus ini, $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial}{\partial x} \left(x\right) & \frac{\partial x}{\partial y}\\\frac{\partial y}{\partial x} & \frac{\partial}{\partial y} \left(y\right)\\\frac{\partial}{\partial x} \left(x y\right) & \frac{\partial}{\partial y} \left(x y\right)\end{array}\right].$$$
Temukan turunan (untuk langkah-langkahnya, lihat kalkulator turunan): $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$
Karena matriks tersebut bukan matriks persegi, determinan Jacobian tidak terdefinisi.
Jawaban
Matriks Jacobian adalah $$$\left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$A.
Determinan Jacobian tidak ada.