Roottorilaskin

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

Määritelmän mukaan $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \nabla\times \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle$$$, tai, yhtäpitävästi, $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\\cos{\left(x y \right)} & e^{x y z} & \sin{\left(x y \right)}\end{array}\right|$$$, missä $$$\times$$$ on ristitulo-operaattori.

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

Laske osittaisderivaatat:

$$$\frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)}$$$ (vaiheista, katso derivointilaskin).

$$$\frac{\partial}{\partial z} \left(e^{x y z}\right) = x y e^{x y z}$$$ (vaiheista, katso derivointilaskin).

$$$\frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) = 0$$$ (vaiheista, katso derivointilaskin).

$$$\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$$$ (vaiheista, katso derivointilaskin).

$$$\frac{\partial}{\partial x} \left(e^{x y z}\right) = y z e^{x y z}$$$ (vaiheista, katso derivointilaskin).

$$$\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$ (vaiheista, katso derivointilaskin).

Nyt sijoita vain löydetyt osittaisderivaatat saadaksesi rotaation: $$$\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle$$$

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