Rotacional de $$$\left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle$$$

Calcule $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle$$$.

Por definição, $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \nabla\times \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle$$$, ou, de forma equivalente, $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\e^{x} \cos{\left(y \right)} & x y^{2} & y z^{3}\end{array}\right|$$$, onde $$$\times$$$ é o operador do produto vetorial.

Logo, $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left\langle \frac{\partial}{\partial y} \left(y z^{3}\right) - \frac{\partial}{\partial z} \left(x y^{2}\right), \frac{\partial}{\partial z} \left(e^{x} \cos{\left(y \right)}\right) - \frac{\partial}{\partial x} \left(y z^{3}\right), \frac{\partial}{\partial x} \left(x y^{2}\right) - \frac{\partial}{\partial y} \left(e^{x} \cos{\left(y \right)}\right)\right\rangle.$$$

Encontre as derivadas parciais:

$$$\frac{\partial}{\partial y} \left(y z^{3}\right) = z^{3}$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial z} \left(x y^{2}\right) = 0$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial z} \left(e^{x} \cos{\left(y \right)}\right) = 0$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial x} \left(y z^{3}\right) = 0$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial x} \left(x y^{2}\right) = y^{2}$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial y} \left(e^{x} \cos{\left(y \right)}\right) = - e^{x} \sin{\left(y \right)}$$$ (para ver os passos, veja a calculadora de derivadas.)

Agora, basta substituir as derivadas parciais encontradas para obter o rotacional: $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left\langle z^{3}, 0, y^{2} + e^{x} \sin{\left(y \right)}\right\rangle$$$

$$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left\langle z^{3}, 0, y^{2} + e^{x} \sin{\left(y \right)}\right\rangle$$$A