7 If F x y z P x y z i Q x y z j R x y z k is a vector field on R and the partial derivatives of P x y z Q x y z and R x y z all exist then the curl of F is the vector field on R defined by OP y and the divergence of F is the function of three variables defined by curl F R 5 y Q R 302 OP 32 i ap Q R r y z Solve the following problems div F x y z j Q 00 k a A vector field F x y z is irrotational if curl F is equal to zero Find constants a b c so that F x 2y az i br 3y z j 4x cy 2z k is irrotational b Show that F 6ry z i 3x z j 3xz y k is irrotational c A vector field F x y z is soleinoidal if its divergence is zero Is F 3y z2 i 4x z j 3r y2 k soleinoidal d For which values of a the vector field F a x y i ay xz j xy 6z k is soleinoidal
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