VC3.21

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\mathrm{curl}f=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ (x+y+1) & 1 & (-x-y) \end{vmatrix}\,

=[\frac{\partial}{\partial y}(-x-y)-\frac{\partial}{\partial z}(1)]i-[\frac{\partial}{\partial x}(-x-y)-\frac{\partial}{\partial z}(x+y+1)]j+[\frac{\partial}{\partial x}(1)-\frac{\partial}{\partial y}(x+y+1)]k\,

=(-1-0)i-(-1-0)j+(0-1)k=-i+j-k\,

Now,f\cdot\mathrm{curl}f=[(x+y+1)i+j+(-x-y)k]\cdot(-i+j-k)\,

=-(x+y+1)+1+(-1)(-x-y)=0\,

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