Matemática, perguntado por christianebspaedfs, 1 ano atrás

Determine o Rotacional do campo vetorial V(x,y,z) = (xz, yz, xy)

a) (x, y - z, 0)
b) (z, y - z, 0)
c) (x, z - y, 0)
d) (y, z - y, 0)
e) (z, z - y, 0)

Soluções para a tarefa

Respondido por silvageeh

Seja F = Pi + Qj + Rk um campo vetorial.

Então, o rotacional de F é calculado por:

$rot F = (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})i+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})j + (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} )k$

Sendo V(x,y,z) = (xz, yz, xy), então:

P = xz
Q = yz
R = xy

Daí, temos que:

$rot V = (\frac{\partial xy }{\partial y}-\frac{\partial yz}{\partial z})i+(\frac{\partial xz}{\partial z}-\frac{\partial xy}{\partial x})j +(\frac{\partial yz}{\partial x}-\frac{\partial xz}{\partial y} )k$
rot V = (x - y)i + (x - y)j + 0k
rot V = (x - y, x - y, 0) → esse é o rotacional de V

Verifique se as alternativas estão corretas.

Respondido por solkarped

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

$\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\textrm{rot}\:\vec{V} = (x - y)\,\vec{i} - (y - x)\,\vec{j} + 0\,\vec{k}\:\:\:}}\end{gathered}$}$

Seja a função vetorial:

$\Large\displaystyle\text{$\begin{gathered} V(x, y, z) = (xz,\,yz,\,xy)\end{gathered}$}$

Organizando o campo vetorial, temos:

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

Sendo V um campo vetorial em R³, podemos dizer que o rotacional de V - denotado por "rot F" - é o produto vetorial entre o operador diferencial e V, isto é:

$\Large\displaystyle\text{$\begin{gathered} \textrm{rot}\:\vec{V} = \nabla\wedge\vec{V}\end{gathered}$}$

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

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

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

$\large\displaystyle\text{$\begin{gathered} = \left(\frac{\partial Z_{V}}{\partial y} - \frac{\partial Y_{V}}{\partial z}\right)\vec{i} - \left(\frac{\partial Z_{V}}{\partial x} - \frac{\partial X_{V}}{\partial z}\right)\vec{j} + \left(\frac{\partial Y_{V}}{\partial x} - \frac{\partial X_{V}}{\partial y}\right)\vec{k}\end{gathered}$}$