Question

Alguém pode me da uma força nessa questão?


Encontre o divergente e o rotacional dos campo vetorial:



V(x, y, z)= 3xi + 2yj - 3zk

Answer 1

Hallemos div V

$\text{div } V=\dfrac{\partial V_1}{\partial x}+\dfrac{\partial V_2}{\partial y}+\dfrac{\partial V_3}{\partial z}\\ \\ \\ \text{div } V=\dfrac{\partial (3x)}{\partial x}+\dfrac{\partial (2y)}{\partial y}+\dfrac{\partial (-3z)}{\partial z}\\ \\ \\ \text{div } V=3+2-3\\ \\ \\ \text{div } V=2$

Hallemos el rotacional

$\text{rot }V=\left| \begin{matrix} i&j&k\\ \partial_x&\partial_y&\partial_z\\ 3x&2y&-3z \end{matrix}\right|\\ \\ \\ \text{rot }V=\left(\dfrac{\partial}{\partial y}(-3z)-\dfrac{\partial}{\partial z}(2y)\right)i-\left(\dfrac{\partial}{\partial x}(-3z)-\dfrac{\partial}{\partial z}(3x)\right)j+\\ \\ +\left(\dfrac{\partial}{\partial x}(2y)-\dfrac{\partial}{\partial y}(3x)\right)k\\ \\ \\ \text{rot }V=0i-0j+0k\\ \\ \\ \text{rot }V=(0,0,0)$