VC3.22

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a). f=(x+3y)i+(y-3z)j+(x-2z)k\,

\mathrm{div}f=\frac{\partial}{\partial x}(x+3y)+\frac{\partial}{\partial y}(y-3z)+\frac{\partial}{\partial z}(x-2z)=1+1-2=0\,

If the divergence of a vector is zero,the vector is a solenoidal vector. Hence,f is a solenoidal vector.

b).Here,\mathrm{div}f=\frac{\partial}{\partial x}(x+3y)+\frac{\partial}{\partial y}(y-2z)+\frac{\partial}{\partial z}(x+az)=0\, [Since f is a solenoidal vector.]

=1+1+a=0\, implies a=-2.

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