Question

Show that x2i +2xyj - 4xzk is solenoidal

Answer 1

Answer :-

$\overrightarrow{V}= x^2i +2xyj - 4xzk$

To prove that V is solenoidal

the condition for this is $\boxed{∇⋅\overrightarrow{V} =0}$

$\implies div.(\overrightarrow{v}) = 0$

$=\frac{∂(\overrightarrow{V})}{∂x} + \frac{∂(\overrightarrow{V})}{∂y} + \frac{∂(\overrightarrow{V})}{∂z}$

$=\frac{∂(x^2)}{∂x} +\frac{∂(2xy)}{∂y} +\frac{∂(-4xz)}{∂z}$

$= 2x +2x - 4x$

= 0

$∇⋅\overrightarrow{V}$ comes out to be zero so ,

$\implies$ given vector is  solenoidal