Question

Encontre o divergente e o rotacional dos campo vetorial:
V(x, y, z) = yi – xj

Answer 1

Dado um campo vetorial $\overrightarrow{\mathbf{V}}:~D\subset \mathbb{R}^3\to \mathbb{R}^3,$

$\overrightarrow{\mathbf{V}}(x,\,y,z)=P\overrightarrow{\mathbf{i}}+Q\overrightarrow{\mathbf{j}}+R\overrightarrow{\mathbf{k}}$

sendo cada uma das funções componentes $P,\,Q,\,R$ funções de três variáveis $x,\,y,\,z.$


Nos pontos do domínio do campo onde as derivadas parciais de $P,\,Q,\,R$ existem, são definidos os seguintes operadores:

• Operador divergente:

$\mathrm{div}\overrightarrow{\mathbf{V}}=\nabla\cdot \overrightarrow{\mathbf{V}}\\\\ \mathrm{div}\overrightarrow{\mathbf{V}}=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$

( o divergente resulta em um campo escalar )


• Operador rotacional:

$\mathrm{rot}\overrightarrow{\mathbf{V}}=\nabla\times \overrightarrow{\mathbf{V}}\\\\\\ \mathrm{rot}\overrightarrow{\mathbf{V}}=\left| \begin{array}{ccc} \overrightarrow{\mathbf{i}}&\overrightarrow{\mathbf{j}}&\overrightarrow{\mathbf{k}}\\\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\\\P&Q&R \end{array} \right|$

$\mathrm{rot}\overrightarrow{\mathbf{V}}=\left(\dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z} \right )\!\overrightarrow{\mathbf{i}}+\left(\dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x} \right )\!\overrightarrow{\mathbf{j}}+\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} \right )\!\overrightarrow{\mathbf{k}}$

( o rotacional resulta em um novo campo vetorial )

______________

Para o campo dado:

$\overrightarrow{\mathbf{V}}(x,\,y,\,z)=y\overrightarrow{\mathbf{i}}-x\overrightarrow{\mathbf{j}}\\\\ \overrightarrow{\mathbf{V}}(x,\,y,\,z)=y\overrightarrow{\mathbf{i}}-x\overrightarrow{\mathbf{j}}+0\overrightarrow{\mathbf{k}}$


As funções componentes são

$\left\{\! \begin{array}{l} P(x,\,y,\,z)=y\\\\ Q(x,\,y,\,z)=-x\\\\ R(x,\,y,\,z)=0 \end{array} \right.$


• O divergente do campo é

$\mathrm{div}\overrightarrow{\mathbf{V}}=\dfrac{\partial}{\partial x}(y)+\dfrac{\partial}{\partial y}(-x)+\dfrac{\partial}{\partial z}(0)\\\\\\ \mathrm{div}\overrightarrow{\mathbf{V}}=0+0+0\\\\ \boxed{\begin{array}{c}\mathrm{div}\overrightarrow{\mathbf{V}}=0 \end{array}}$

( função nula )


• O rotacional do campo é

$\mathrm{rot}\overrightarrow{\mathbf{V}}=\left| \begin{array}{ccc} \overrightarrow{\mathbf{i}}&\overrightarrow{\mathbf{j}}&\overrightarrow{\mathbf{k}}\\\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\\\y&-x&0 \end{array} \right|\\\\\\\\ \mathrm{rot}\overrightarrow{\mathbf{V}}=\left(\dfrac{\partial}{\partial y}(0)-\dfrac{\partial }{\partial z}(-x) \right )\!\overrightarrow{\mathbf{i}}+\left(\dfrac{\partial}{\partial z}(y)-\dfrac{\partial}{\partial x}(0) \right )\!\overrightarrow{\mathbf{j}}+\left(\dfrac{\partial}{\partial x}(-x)-\dfrac{\partial}{\partial y}(y) \right )\!\overrightarrow{\mathbf{k}}$


$\mathrm{rot}\overrightarrow{\mathbf{V}}=(0-0)\overrightarrow{\mathbf{i}}+(0-0)\overrightarrow{\mathbf{j}}+(-1-1)\overrightarrow{\mathbf{k}}\\\\\\ \boxed{\begin{array}{c}\mathrm{rot}\overrightarrow{\mathbf{V}}=-2\overrightarrow{\mathbf{k}} \end{array}}$

( campo vetorial que só tem a 3ª componente não-nula )


Bons estudos! :-)