Vtyjkyuk

$(3x^2 +6y)dx -14yzdy +20xz^2$ is an exact differential then why the curl of $(3x^2+6y) hati -14yzhatj +20xz^2hatk $ is not zero ? If you experiment if the differential equation $$partial ^2 (3x^2 +6y) / partial y partial z$$ = $$partial ^2 -14yz/partial zpartial x$$= $$partial ^2 20xz^2 /partial x partial y$$=0 but the curl of $$(3x^2+6y) hati -14yzhatj +20xz^2hatk $$ is not zero. Now if the differential equation is exact then the vector should be a gradient whose curl should be zero . Where am I wrong?

The curl equals $(14y, -20 z^2, -6) =left(fracpartial F_zpartial y - fracpartial F_ypartial zright) mathbfi + left(fracpartial F_xpartial z - fracpartial F_zpartial x right) mathbfj + left(fracpartial F_ypartial x - fracpartial F_xpartial y right) mathbfk = beginbmatrixfracpartial F_zpartial y - fracpartial F_ypartial z \ fracpartial F_xpartial z - fracpartial F_zpartial x \ fracpartial F_ypartial x - fracpartial F_xpartial yendbmatrix$ There is no second order differential unlike what you posted.