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Kalkulator Jakobian
Hitung Jakobian langkah demi langkah
Kalkulator akan menghitung matriks Jacobian dari himpunan fungsi dan determinan Jacobian (jika memungkinkan), dengan menampilkan langkah-langkahnya.
Masukan Anda
Hitung Jakobian dari $$$\left\{x = r \cos{\left(\theta \right)}, y = r \sin{\left(\theta \right)}\right\}$$$.
Solusi
Matriks Jacobian didefinisikan sebagai berikut: $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{array}\right].$$$
Dalam kasus ini, $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\frac{\partial}{\partial r} \left(r \cos{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \cos{\left(\theta \right)}\right)\\\frac{\partial}{\partial r} \left(r \sin{\left(\theta \right)}\right) & \frac{\partial}{\partial \theta} \left(r \sin{\left(\theta \right)}\right)\end{array}\right].$$$
Temukan turunan (untuk langkah-langkahnya, lihat kalkulator turunan): $$$J{\left(x,y \right)}\left(r, \theta\right) = \left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right]$$$
Determinan Jacobian adalah determinan dari matriks Jacobian: $$$\left|\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right| = r$$$ (untuk langkah-langkahnya, lihat kalkulator determinan).
Jawaban
Matriks Jacobian adalah $$$\left[\begin{array}{cc}\cos{\left(\theta \right)} & - r \sin{\left(\theta \right)}\\\sin{\left(\theta \right)} & r \cos{\left(\theta \right)}\end{array}\right]$$$A.
Determinan Jacobian adalah $$$r$$$A.