Curl Calculator

The calculator will find the curl of the given vector field, with steps shown.

Related calculators: Partial Derivative Calculator, Cross Product Calculator, Matrix Determinant Calculator

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Your Input

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

Solution

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

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

Find the partial derivatives:

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

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

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

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

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

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

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

Answer

$$$\text{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle$$$A