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

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

By definition, $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \nabla\times \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle$$$, or, equivalently, $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\e^{x} \cos{\left(y \right)} & x y^{2} & y z^{3}\end{array}\right|$$$, where $$$\times$$$ is the cross product operator.

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

Find the partial derivatives:

$$$\frac{\partial}{\partial y} \left(y z^{3}\right) = z^{3}$$$ (for steps, see derivative calculator).

$$$\frac{\partial}{\partial z} \left(x y^{2}\right) = 0$$$ (for steps, see derivative calculator).

$$$\frac{\partial}{\partial z} \left(e^{x} \cos{\left(y \right)}\right) = 0$$$ (for steps, see derivative calculator).

$$$\frac{\partial}{\partial x} \left(y z^{3}\right) = 0$$$ (for steps, see derivative calculator).

$$$\frac{\partial}{\partial x} \left(x y^{2}\right) = y^{2}$$$ (for steps, see derivative calculator).

$$$\frac{\partial}{\partial y} \left(e^{x} \cos{\left(y \right)}\right) = - e^{x} \sin{\left(y \right)}$$$ (for steps, see derivative calculator).

Now, just plug in the found partial derivatives to get the curl: $$$\operatorname{curl} \left\langle e^{x} \cos{\left(y \right)}, x y^{2}, y z^{3}\right\rangle = \left\langle z^{3}, 0, y^{2} + e^{x} \sin{\left(y \right)}\right\rangle.$$$

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