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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