A two$-$dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region.

a$.$ Verify that the curl and divergence of the given field is zero.

b$.$ Find a potential function $$ and a stream function $$ for the field.

c$.$ Verify that $$ and $$ satisfy Laplace's equation ${}_{}+{}_{yy}={}_{}+{}_{yy}=0.$

$F=($:$6x^3-18x{y}^2,6y^3-18x^{2}y$:$)$

a$.$ Verify that the given vector field has zero curl.

A$.$ The sum of the partial derivatives, $f_x$ and $g_y,$ equals zero.

B$.$ The difference of the partial derivatives, $f_x$ and $g_y,$ equals zero.

C$.$ The difference of the partial derivatives, $g_x$ and $f_y,$ equals zero.

D$.$ The sum of the partial derivatives, $g_x$ and $f_y,$ equals zero.