旋度计算器

计算 $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle$$$。

根据定义，$$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \nabla\times \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle$$$，或等价地，$$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\\cos{\left(x y \right)} & e^{x y z} & \sin{\left(x y \right)}\end{array}\right|$$$，其中 $$$\times$$$ 是叉乘运算符。

因此，$$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle \frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) - \frac{\partial}{\partial z} \left(e^{x y z}\right), \frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) - \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right), \frac{\partial}{\partial x} \left(e^{x y z}\right) - \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right)\right\rangle$$$。

求偏导数：

$$$\frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)}$$$（步骤参见导数计算器）。

$$$\frac{\partial}{\partial z} \left(e^{x y z}\right) = x y e^{x y z}$$$（步骤参见导数计算器）。

$$$\frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) = 0$$$（步骤参见导数计算器）。

$$$\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$$$（步骤参见导数计算器）。

$$$\frac{\partial}{\partial x} \left(e^{x y z}\right) = y z e^{x y z}$$$（步骤参见导数计算器）。

$$$\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$（步骤参见导数计算器）。

现在，只需将求得的偏导数代入即可得到旋度：$$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle$$$

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