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Let

$F(x_1, x_2, x_3) = F_1(x_1, x_2, x_3) i + F_2(x_1, x_2, x_3)j + F_3(x_1, x_2, x_3)k$

be a vector field. If $F_i$ is not a a function of $x_i$ for all $i$, then $F$ is incompressible. If $F_i$ is only a function of $x_i$ for all $i$, then $F$ is irrotational. However, we can only apply one of these tests at a time.

Are there any "quick" (no computation or plotting allowed) techniques for telling when a vector field must be BOTH irrotational and incompressible?

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If you want neither computation (say, of the curl and divergence...) nor plotting, then what the deuce do you want? –  Henning Makholm Sep 28 '11 at 2:37
    
Am I missed something? If $F_i$ is not a function of $x_i$ and only a function of $x_i$, then it's constant. –  Bill Cook Sep 28 '11 at 3:02
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