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Kalkulator Rotasi
Hitung rotasi langkah demi langkah
Kalkulator akan menghitung rotasi dari medan vektor yang diberikan, dengan menampilkan langkah-langkahnya.
Kalkulator terkait: Kalkulator Turunan Parsial, Kalkulator Hasil Kali Silang, Kalkulator Determinan Matriks
Masukan Anda
Hitung $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle$$$.
Solusi
Menurut definisi, $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \nabla\times \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle$$$, atau, secara ekuivalen, $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\\cos{\left(x y \right)} & e^{x y z} & \sin{\left(x y \right)}\end{array}\right|$$$, di mana $$$\times$$$ adalah operator hasil kali silang.
Dengan demikian, $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle \frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) - \frac{\partial}{\partial z} \left(e^{x y z}\right), \frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) - \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right), \frac{\partial}{\partial x} \left(e^{x y z}\right) - \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right)\right\rangle.$$$
Tentukan turunan parsial:
$$$\frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)}$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
$$$\frac{\partial}{\partial z} \left(e^{x y z}\right) = x y e^{x y z}$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
$$$\frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) = 0$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
$$$\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
$$$\frac{\partial}{\partial x} \left(e^{x y z}\right) = y z e^{x y z}$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
$$$\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).
Sekarang, cukup masukkan turunan parsial yang telah ditemukan untuk mendapatkan rotasi (curl): $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle$$$
Jawaban
$$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle$$$A