Question

Qual o divergente da função f(x, y, z) = (3y5z7, - 7x2z2 , 5 x y3) ?​​​​​​​

A.
y5z7 - x2z2 + 5 x y3


B.
105y4z6 - 4xz + 15x y2


C.
0


D.
3y5z7 i - 7x2z2 j + 5 x y3k


E.
z7 - 7x2 + 5y3

Answer 1

$\Large\displaystyle\begin{gathered}\vec{F}(x,y,z)=\left(3y5z^7\right)\vec{i} +\left(-7x^2z^2\right)\vec{j}+\left( 5 x y^3\right)\vec{k} \end{gathered}$

$\Large\displaystyle\begin{gathered}\nabla\cdot \vec{F}=\frac{\partial F_x}{\partial x}\ \vec{i}+ \frac{\partial F_y}{\partial y}\ \vec{j} +\frac{\partial F_z}{\partial z}\ \vec{k}\end{gathered}$

$\Large\displaystyle\begin{gathered}\nabla\cdot \vec{F}=\frac{\partial}{\partial x}\left(3y^5z^7\right)\ \vec{i}+ \frac{\partial }{\partial y}\left(-7x^2z^2\right)\ \vec{j} +\frac{\partial}{\partial z}\left(5xy^3\right)\ \vec{k}\end{gathered}$

$\Large\displaystyle\begin{gathered}\nabla\cdot \vec{F}= 0+0 +0\end{gathered}$

$\Large\displaystyle\text{ \begin{gathered}\boxed{\nabla\cdot \vec{F}= 0}\end{gathered} }$

Answer 2

✅ Após resolver os cálculos, concluímos que o divergente da referida função vetorial é:

                          $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\ \textrm{div}\:\vec{F}= 0\:\:\:}}\end{gathered} }$  

Portanto, a opção correta é:

                     $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf Alternativa\:C\:\:\:}}\end{gathered} }$

               

Seja a função vetorial dada:

            $\Large\displaystyle\begin{gathered} f(x, y, z) = (3y^{5}z^{7},\,-7x^{2} z^{2},\,5xy^{3})\end{gathered}$

Organizando a função vetorial, temos:

   $\Large\displaystyle\begin{gathered}\vec{F}(x, y, z) = (3y^{5}z^{7})\vec{i} + (-7x^{2}z^{2})\vec{j} + (5xy^{3})\vec{k}\end{gathered}$

O divergente de uma função "F" vetorial no R³ - definido por "div F" - é o campo escalar dado pelo produto escalar do operador diferencial com a função F.  Ou seja

      $\Large\displaystyle\begin{gathered} \textrm{div}\:\vec{F} = \nabla\cdot\vec{F}\end{gathered}$      

                    $\Large\displaystyle\begin{gathered} = \bigg(\frac{\partial}{\partial x},\,\frac{\partial}{\partial y},\,\frac{\partial}{\partial z}\bigg)\cdot(X_{F}\vec{i},\:Y_{F}\vec{j},\,Z_{F}\vec{k})\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = \frac{\partial}{\partial x} \cdot X_{F}\vec{i} +\,\frac{\partial}{\partial y}\cdot Y_{F}\vec{j} +\,\frac{\partial}{\partial z}\cdot Z_{F}\vec{k}\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = \frac{\partial}{\partial x}\:(3y^{5} z^{7})\:\vec{i} + \frac{\partial}{\partial y}\:(-7x^{2} z^{2})\:\vec{j} + \frac{\partial}{\partial z}\:(5xy^{3})\:\vec{k}\end{gathered}$

                     $\Large\displaystyle\begin{gathered} = 0\:\vec{i} + 0\:\vec{j} + 0\:\vec{k}\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = 0 + 0 + 0\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = 0\end{gathered}$

Portanto, o divergente procurado é:

        $\Large\displaystyle\begin{gathered} \textrm{div}\:\vec{F}= 0 \end{gathered}$

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2. https://brainly.com.br/tarefa/1295523
3. https://brainly.com.br/tarefa/1297394
4. https://brainly.com.br/tarefa/45059846