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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.

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Your results are correct, but the Laplace equation is indeed satisfied:

$$ \begin{eqnarray} \Delta F_x&=&\Delta(ax^2+cy^2)=2a+2c=0\;,\\ \Delta F_y&=&\Delta(x^2-y^2)=2-2=0\;,\\ \Delta F_z&=&\Delta(2y+2z)=0\;. \end{eqnarray} $$

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Perfect Have my head around it now, many thanks. – user24930 Mar 27 '12 at 19:15

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