Question

Ensino superior calculo ajuda
Encontre rot F e div F, onde

F(x; y; z) = xyzi - x^2yk.

Answer 1

Sobre campos vetoriais cujas derivadas parciais das funções componentes existam, definimos os operadores diferenciais rotacional e divergente.


$\mathtt{\overrightarrow{\mathtt{F}}(x,\,y,\,z)=P\overrightarrow{\mathtt{i}}+Q\overrightarrow{\mathtt{j}}+R\overrightarrow{\mathtt{k}}}\\\\ \mathtt{\overrightarrow{\mathtt{F}}(x,\,y,\,z)=xyz\overrightarrow{\mathtt{i}}-x^2y \overrightarrow{\mathtt{k}}}\\\\ \mathtt{\overrightarrow{\mathtt{F}}(x,\,y,\,z)=xyz\overrightarrow{\mathtt{i}}+0\overrightarrow{\mathtt{j}}-x^2y \overrightarrow{\mathtt{k}}}$

_________

•   O rotacional de $\mathtt{\overrightarrow{\mathtt{F}}}:$

$\mathtt{rot}\overrightarrow{\mathtt{F}}\mathtt{=\nabla\times \overrightarrow{\mathtt{F}}}\\\\\\ =\left|\begin{array}{ccc} \overrightarrow{\mathtt{i}}&\overrightarrow{\mathtt{j}}&\overrightarrow{\mathtt{k}}\\\\ \mathtt{\frac{\partial}{\partial x}}&\mathtt{\frac{\partial}{\partial y}}&\mathtt{\frac{\partial}{\partial z}}\\\\ \mathtt{P}&\mathtt{Q}&\mathtt{R} \end{array}\right|\\\\\\\\ =\left|\begin{array}{ccc} \overrightarrow{\mathtt{i}}&\overrightarrow{\mathtt{j}}&\overrightarrow{\mathtt{k}}\\\\ \mathtt{\frac{\partial}{\partial x}}&\mathtt{\frac{\partial}{\partial y}}&\mathtt{\frac{\partial}{\partial z}}\\\\ \mathtt{xyz}&\mathtt{0}&\mathtt{-x^2 y} \end{array}\right|$


$=\mathtt{\left(\dfrac{\partial}{\partial y}(-x^2 y)-\dfrac{\partial}{\partial z}(0)\right)\!\!\overrightarrow{\mathtt{i}}+\left(\dfrac{\partial}{\partial z}(xyz)-\dfrac{\partial}{\partial x}(-x^2 y)\right)\!\!\overrightarrow{\mathtt{j}}+\left(\dfrac{\partial}{\partial x}(0)-\dfrac{\partial}{\partial y}(xyz)\right)\!\!\overrightarrow{\mathtt{k}}}$

$=\mathtt{(-x^2-0)\overrightarrow{\mathtt{i}}+(xy-(-2xy))\overrightarrow{\mathtt{j}}+(0-xz)\overrightarrow{\mathtt{k}}}\\\\ =\boxed{\begin{array}{c}\mathtt{-x^2\overrightarrow{\mathtt{i}}+3xy\overrightarrow{\mathtt{j}}-xz\overrightarrow{\mathtt{k}}} \end{array}}$


(o rotacional nos fornece um novo campo vetorial)

_________

•   O divergente de $\mathtt{\overrightarrow{\mathtt{F}}}:$

$\mathtt{div\overrightarrow{\mathtt{F}}=\nabla \cdot \overrightarrow{\mathtt{F}}}\\\\ =\mathtt{\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}}\\\\\\ =\mathtt{\dfrac{\partial}{\partial x}(xyz)+\dfrac{\partial}{\partial y}(0)+\dfrac{\partial}{\partial z}(-x^2 y)}\\\\\\ =\mathtt{yz+0+0}\\\\\\ =\boxed{\begin{array}{c}\mathtt{yz} \end{array}}$


(o divergente nos fornece um campo escalar – função real)


Dúvidas? Comente.


Bons estudos! :-)