Question 26.6: Show that the vector field v = x sin yi+y sin xj−z(sin x+sin......

We have

v_{x}=x\sin y\,              so that        {\frac{\partial v_{x}}{\partial x}}=\sin y\,

Also,

v_{\mathrm{y}}=y\sin x                so that         {\frac{\partial v_{\mathrm{y}}}{\partial y}}=\sin x\,

Finally,

v_{z}=-z(\sin x+\sin y)                so that              {\frac{\partial v_{z}}{\partial z}}=-(\sin x+\sin y)

Therefore,

and hence v is solenoidal.