Question

Encontre o divergente e o rotacional do campo vetorial:
V(x, y, z) = 3xi + 2yj – 3zk

Answer 1

Temos um campo vetorial $\overrightarrow{\mathbf{V}}:~\mathbb{R}^3\to\mathbb{R}^3$

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

onde cada uma das componentes $P,\,Q,\,R$ são funções de 3 variáveis $x,\,y,\,z.$


Nos pontos onde as derivadas parciais das componentes $P,\,Q,\,R$ existirem, define-se os seguintes operadores diferenciais sobre o campo vetorial $\overrightarrow{\mathbf{V}}:$


• Operador divergente:

$\mathrm{div}\overrightarrow{\mathbf{V}}(x,\,y,\,z)=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$


Podemos usar também o operador nabla, e expressar o divergente de forma semelhante ao produto escalar de dois vetores:

$\mathrm{div}\overrightarrow{\mathbf{V}}=\nabla\cdot \overrightarrow{\mathbf{V}}=\left(\dfrac{\partial}{\partial x},\,\dfrac{\partial}{\partial y},\,\dfrac{\partial}{\partial z}\right)\cdot (P,\,Q,\,R)$


O operador divergente de um campo vetorial nos fornece um campo escalar.


• Operador rotacional:

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


Podemos usar também o operador nabla, e expressar o rotacional de forma semelhante ao produto vetorial:

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


O rotacional de um campo vetorial nos fornece um novo campo vetorial.

____________

Para o enunciado dado temos

$\overrightarrow{\mathbf{V}}(x,\,y,\,z)=3x\overrightarrow{\mathbf{i}}+2y\overrightarrow{\mathbf{j}}-3z\overrightarrow{\mathbf{k}}$


Portanto, as funções componentes são

$\left\{ \!\begin{array}{l} P(x,\,y,\,z)=3x\\\\ Q(x,\,y,\,z)=2y\\\\ R(x,\,y,\,z)=-3z \end{array} \right.$


• O divergente do campo é

$\mathrm{div} \overrightarrow{\mathbf{V}}=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}\\\\\\ \mathrm{div} \overrightarrow{\mathbf{V}}=\dfrac{\partial}{\partial x}(3x)+\dfrac{\partial}{\partial y}(2y)+\dfrac{\partial}{\partial z}(-3z)\\\\\\ \mathrm{div} \overrightarrow{\mathbf{V}}=3+2-3\\\\ \therefore~~\boxed{\begin{array}{c}\mathrm{div} \overrightarrow{\mathbf{V}}=2 \end{array}}$

( o divergente é uma função constante )


• O rotacional do campo é

$\mathrm{rot} \overrightarrow{\mathbf{V}}=\left(\dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z} \right )\overrightarrow{\mathbf{i}}+\left(\dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x} \right )\overrightarrow{\mathbf{j}}+\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} \right )\overrightarrow{\mathbf{k}}\\\\\\ \mathrm{rot} \overrightarrow{\mathbf{V}}=\left(\dfrac{\partial}{\partial y}(-3z)-\dfrac{\partial}{\partial z}(2y) \right )\overrightarrow{\mathbf{i}}+\left(\dfrac{\partial}{\partial z}(3x)-\dfrac{\partial}{\partial x}(-3z) \right )\overrightarrow{\mathbf{j}}+\left(\dfrac{\partial}{\partial x}(2y)-\dfrac{\partial}{\partial y}(3x) \right )\overrightarrow{\mathbf{k}}$

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

( o rotacional é o campo vetorial nulo )


Bons estudos! :-)