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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.

Dipisahkan dengan koma.
Biarkan kosong untuk deteksi otomatis atau tentukan variabel seperti x,y (dipisahkan dengan koma).

Jika kalkulator tidak menghitung sesuatu atau Anda menemukan kesalahan, atau Anda memiliki saran/masukan, silakan hubungi kami.

Masukan Anda

Hitung Jakobian dari $$$\left\{x = r \cos{\left(\tanh{\left(\eta \right)} \right)}, y = r \sin{\left(\tanh{\left(\eta \right)} \right)}\right\}$$$.

Solusi

Matriks Jacobian didefinisikan sebagai berikut: $$$J{\left(x,y \right)}\left(\eta, r\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial \eta} & \frac{\partial x}{\partial r}\\\frac{\partial y}{\partial \eta} & \frac{\partial y}{\partial r}\end{array}\right].$$$

Dalam kasus ini, $$$J{\left(x,y \right)}\left(\eta, r\right) = \left[\begin{array}{cc}\frac{\partial}{\partial \eta} \left(r \cos{\left(\tanh{\left(\eta \right)} \right)}\right) & \frac{\partial}{\partial r} \left(r \cos{\left(\tanh{\left(\eta \right)} \right)}\right)\\\frac{\partial}{\partial \eta} \left(r \sin{\left(\tanh{\left(\eta \right)} \right)}\right) & \frac{\partial}{\partial r} \left(r \sin{\left(\tanh{\left(\eta \right)} \right)}\right)\end{array}\right].$$$

Temukan turunan (untuk langkah-langkahnya, lihat kalkulator turunan): $$$J{\left(x,y \right)}\left(\eta, r\right) = \left[\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right]$$$

Determinan Jacobian adalah determinan dari matriks Jacobian: $$$\left|\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right| = - r \operatorname{sech}^{2}{\left(\eta \right)}$$$ (untuk langkah-langkahnya, lihat kalkulator determinan).

Jawaban

Matriks Jacobian adalah $$$\left[\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right].$$$A

Determinan Jacobian adalah $$$- r \operatorname{sech}^{2}{\left(\eta \right)}$$$A.