Question

calcular o divergente de x^2yz+xy^2z+xyz^2​

Answer 1

F (x, y, z) = x²yz + xy²z + xyz²​

div F = [d (x²yz) / dx] + [d (xy²z) / dy] + [d (xyz²​) / dy]

div F = 2xyz + 2xyz + 2xyz

div F = 6xyz

Answer 2

✅ Após resolver os cálculos, concluímos que o divergente do referido campo vetorial é:

                 $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\ \textrm{div}\:\vec{V}= 6xyz\:\:\:}}\end{gathered} }$

Seja o dado:

                   $\Large\displaystyle\begin{gathered} x^{2}yz + xy^{2}z + xyz^{2}\end{gathered}$

Organizando o campo vetorial "V" temos:

     $\Large\displaystyle\begin{gathered} \vec{V}(x, y, z) = (x^{2}yz)\,\vec{i} + (xy^{2}z)\,\vec{j} + (xyz^{2})\,\vec{k}\end{gathered}$

O divergente de um campo "V" vetorial no V³ - definido por "div V" - é o campo escalar dado pelo produto escalar do operador diferencial com o campo  V.  Ou seja

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

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

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

                      $\Large\displaystyle\begin{gathered} = \frac{\partial}{\partial x}(x^{2}yz)\,\vec{i} + \frac{\partial}{\partial y}(xy^{2}z)\,\vec{j} + \frac{\partial}{\partial z}(xyz^{2})\,\vec{k}\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = 2xyz\,\vec{i} + x2yz\,\vec{j} + xy2z\,\vec{k}\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = 2xyz + 2xyz + 2xyz\end{gathered}$

                      $\Large\displaystyle\begin{gathered} = 6xyz\end{gathered}$

✅ Portanto o divergente é:

        $\Large\displaystyle\begin{gathered} \textrm{div}\:\vec{V}= 6xyz\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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