I have a straight forward divergence and curl question
Let F be the vector field
$F=(ax^{2}+bxy+cy^{2}-2x)i + (x^{2}+xy-y^{2}+bz)j+ (2y+2z)k$
Determine a,b,c since the vector field is solenodial, hence for this I have a=-0.5, b=2 and $c\in\mathbb{R}$
Now dertermine a,b,c since the vector Field is irrptational, hence for this I have, b=2, c=0.5 and now $a\in\mathbb{R}$.
Lastly fix the values of a,b,c such that the vector field f is both solenodial and irrotational and check that the vector field satisfies the laplace equation. However for this I have choosen my a=-0.5, b=2 and c=0.5, then when computing the laplace equation $\neq 0$, I must have gone wrong somewhere, any help would be most appreciated, many thanks.