Question

Ache o divergente e o rotacional do campo vetorial:
V(x, y, z) = zi + xj + yk

Answer 1

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


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O divergente do campo é

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


( o divergente é nulo, então o campo $\overrightarrow{\mathbf{V}}$
 é incompressível )

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O rotacional do campo é

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

$\mathrm{rot}\overrightarrow{\mathbf{V}}=\left(\dfrac{\partial}{\partial y}(y)-\dfrac{\partial}{\partial z}(x)\right)\overrightarrow{\mathbf{i}}+\left(\dfrac{\partial}{\partial z}(z)-\dfrac{\partial}{\partial x}(y)\right)\overrightarrow{\mathbf{j}}+\left(\dfrac{\partial}{\partial x}(x)-\dfrac{\partial}{\partial y}(z)\right)\overrightarrow{\mathbf{k}}\\\\\\ \mathrm{rot}\overrightarrow{\mathbf{V}}=(1-0)\overrightarrow{\mathbf{i}}+(1-0)\overrightarrow{\mathbf{j}}+(1-0)\overrightarrow{\mathbf{k}}$

$\boxed{\begin{array}{c}\mathrm{rot}\overrightarrow{\mathbf{V}}=\overrightarrow{\mathbf{i}}+\overrightarrow{\mathbf{j}}+\overrightarrow{\mathbf{k}} \end{array}}$


Bons estudos! :-)