Vektorikentän $$$\left\langle y z, x z, x y\right\rangle$$$ rotaatio

Laske $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle$$$.

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

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

Laske osittaisderivaatat:

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

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

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

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

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

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

Nyt sijoita vain löydetyt osittaisderivaatat saadaksesi rotaation: $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$

$$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$A