with(linalg):
F:=vector([12*x*y+y*z,6*x^2+x*z,x*y]);
[ 2 ]
F := [12 x y + y z, 6 x + x z, x y]
curl(F,[x,y,z]);
[0, 0, 0]
Therefore the vector field is conservative.
F:=vector([z*exp(x*z),0,x*exp(x*z)]);
F := [z exp(x z), 0, x exp(x z)]
curl(F,[x,y,z]);
[0, 0, 0]
Therefore the vector field is conservative.
F:=vector([sin(x),y^2,exp(z)]);
[ 2 ]
F := [sin(x), y , exp(z)]
curl(F,[x,y,z]);
[0, 0, 0]
Therefore the vector field is conservative.
F:=vector([3*x^2*y*z^2,x^3*z^2,x^3*y*z]);
[ 2 2 3 2 3 ]
F := [3 x y z , x z , x y z]
curl(F,[x,y,z]);
[ 3 2 ]
[-x z, 3 x y z, 0]
Therefore the vector field is not conservative.
F:=vector([2*x/(x^2+y^2),2*y/(x^2+y^2),2*z]);
[ x y ]
F := [2 -------, 2 -------, 2 z]
[ 2 2 2 2 ]
[ x + y x + y ]
curl(F,[x,y,z]);
[0, 0, 0]
In this case, the curl is zero, but that doesn't
tell us anything.