Divergenssilaskin

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

Määritelmän mukaan $$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \nabla\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$, tai yhtäpitävästi $$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$, missä $$$\cdot$$$ on skalaaritulo-operaattori.

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

Laske komponentin 1 osittaisderivaatta muuttujan $$$x$$$ suhteen: $$$\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$$$ (vaiheet: katso derivative calculator).

Laske komponentin 2 osittaisderivaatta muuttujan $$$y$$$ suhteen: $$$\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$ (vaiheet: katso derivative calculator).

Laske komponentin 3 osittaisderivaatta muuttujan $$$z$$$ suhteen: $$$\frac{\partial}{\partial z} \left(e^{z}\right) = e^{z}$$$ (vaiheet: katso derivative calculator).

Laske nyt vain yllä olevat lausekkeet yhteen saadaksesi divergenssin: $$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = - x \sin{\left(x y \right)} + y \cos{\left(x y \right)} + e^{z}.$$$

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