Question

Find 'a' such that
(3x - 2y +z)i +(4x + ay – z)]+(x -y +2z)k
is solenoidal
a = -5

a= -2
a = -3
a = 4​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Given a vector , find 'a' such that the vector is solenoidal

Explanation:

1. A vector is said to be solenoidal at a point if Divergence of the vector is zero at the point. But the vector may not be solenoidal at other points of region under consideration.
2. divergence of a vector $v=P(i)+Q(j)+R(k)$ is given by ,                                      $div.\ v= \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R }{\partial z}$    
3. we have ,                                $v=(3x - 2y +z)i +(4x +ay-z)j+(x -y +2z)k\\P=(3x - 2y +z),\ Q=(4x +ay-z),\ R=(x -y +2z)\\\\\frac{\partial P}{\partial x}=\frac{\partial (3x - 2y +z) }{\partial x}= 3 \\\frac{\partial Q}{\partial y}=\frac{\partial (4x +ay-z)}{\partial y}= a \\\frac{\partial R}{\partial z}=\frac{\partial (x-y +2z) }{\partial z}= 2$    
4. now for this vector to be solenoidal, $div.\ v=0$  hence,                                         $\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R }{\partial z}=0 \\3+a+2=0\\a=-5$----->ANSWER