Question #2184

Determine if the following vector is solenoidal,
A = 3y^4 z^2 i + 4x^3 z^2j- 3x^2y^2

Expert's answer

Let's find the divergence of A:

<img src="/cgi-bin/mimetex.cgi?%5Cnabla%20%5Ccdot%20%5Cvec%7BA%7D%20=%20%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%283y%5E4z%5E2%29+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%284x%5E3z%5E2%29%20+%20%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20z%7D%28-3x%5E2y%5E2%29%20=%200+0+0%20=%200" title="\nabla \cdot \vec{A} = \frac{\partial}{\partial x}(3y^4z^2)+\frac{\partial}{\partial y}(4x^3z^2) + \frac{\partial}{\partial z}(-3x^2y^2) = 0+0+0 = 0">

The vector A is solenoidal.

<img src="/cgi-bin/mimetex.cgi?%5Cnabla%20%5Ccdot%20%5Cvec%7BA%7D%20=%20%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%283y%5E4z%5E2%29+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%284x%5E3z%5E2%29%20+%20%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20z%7D%28-3x%5E2y%5E2%29%20=%200+0+0%20=%200" title="\nabla \cdot \vec{A} = \frac{\partial}{\partial x}(3y^4z^2)+\frac{\partial}{\partial y}(4x^3z^2) + \frac{\partial}{\partial z}(-3x^2y^2) = 0+0+0 = 0">

The vector A is solenoidal.

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